
Book *~Bf& 

OopightN 

COPYRIGHT DEPOSIT. 



Benjamin's Machine Design. 

By Charles H. Benjamin, Professor 
in the Case School of Science. 

He-skins' Hydraulics. 

By L. M. Hoskins, Professor in Leland 
Stanford University. 

Adams' Alternating-current 
Machines. 

By C. A. Adams, Professor in Harvard 
University. [In preparation. ] 



HENRY HOLT AND COMPANY 

NEW YORK CHICAGO 



MACHINE DESIGN 



BY 



CHARLES H. BENJAMIN 

Professor of Mechanical Engineering in the 
Case School of Applied Science 




NEW YORK 
HENRY HOLT AND COMPANY 

1906 



LIBRARY of CONGRESS 
Two Copies Received 

DEC 1 1906 

^ Copyright Entry 
CLASS A XXC, No. 



i <* f 

COPY B. 






Copyright, 1906 

BY 

HENRY HOLT AND COMPANY 



I: 






PREFACE. 



This book embodies to a considerable extent the 
writer's experience in teaching and in commercial 
work. While the underlying mechanical principles of 
machine design are permanent, the application of them 
is continually changing. The researches of the ex- 
perimenter and the practice of the builder are always 
showing good reasons for the modification of design* 

Although the present work was prepared primarily 
for a text-book, it contains mainly what the writer has 
found necessary in his own practice as an engineer. 
As far as possible the formulas for the strength and 
stiffness of machine details have been fortified by the 
results of experiments or by the practical experience of 
manufacturers. 

Attention is called particularly to the experiments 
on cast-iron cylinders, pipe fittings, helical springs, 
roller bearings, gear teeth, pulley arms, and the 
bursting strength of fly-wheels. 

What the student needs to learn before graduation 
is also what he needs to remember after, and it is 
hoped that this book contains the necessary facts and 
principles and not too much else. 

C. H. B. 



in 



TABLE OF CONTENTS. 



CHAPTER. PAGE. 

I. Units and Tables 1 

I. Units. 2. Abbreviations. 3. Materials. 4. Notation. 
5. Formulas. 6. Profiles of uniform strength. 7. Factors 
of safety. 

II. Frame Design 15 

8. General principles of design. 9. Machine supports. 10. 
Machine frames. 

III. Cylinders and Pipes 25 

II. Thin shells. 12. Thick shells. 13. Steel and wrought . 
iron pipe. 14. Strength of boiler tubes. 15. Pipe fittings. 
16. Steam cylinders. 17. Thickness of flat plates. 

IV. Fastenings 54 

18. Bolts and nuts. 19. Machine screws. 20. Eye bolts 
and hooks. 21. Riveted joints. 22. Lap joints. 23. Butt 
joints with two straps. 24. Efficiency of joints. 25. Butt 
joints with unequal straps. 26. Practical rules. 27. Riv- 
eted joints for narrow plates. 28. Joint pins. 29. Cotters. 

V. Springs 73 

30. Helical springs. 31. Square wire. 32. Experiments. 

33. Springs in torsion. 34. Flat springs. 35. Elliptic and 
semi-elliptic springs. 

VI. Sliding Bearings 86 

36. Slides in general. 37. Angular slides. 38. Gibbed 
slides. 39. Flat slides. 40. Circular guides. 41. Stuffing 
boxes. 

VII. Journals, Pivots and Bearings 96 

42. Journals. 43. Adjustment. 44. Lubrication. 45. Fric- 
tion of journals. 46. Limits of pressure. 47. Heating of 
journals. 48. Experiments. 49. Strength and stiffness of 
journals. 50. Caps and bolts. 51. Step bearings. 52. 
Friction of pivots. 53. Flat collar. 54. Conical pivot. 55. 
Schiele's pivot. 56. Multiple bearing. 

V 



vi MACHINE DESIGN. 

CHAPTER. PAGE. 

VIII. Ball and Roller Bearings 118 

57. General principles. 58. Journal bearings. 59. Step 
bearings. 60. Materials and wear. 61. Design of bearings. 
62. Roller bearings. 63. Grant roller bearing. 64. Hyatt 
rollers. 65. Roller step bearings. 

IX. Shafting, Couplings and Hangers 130 

66. Strength of shafting. 67. Couplings. 68. Clutches. 

69. Coupling bolts. 70. Shafting keys. 71. Hangers and 
boxes. 

X. Gears, Pulleys and Cranks 146 

72. Gear teeth. 73. Strength of teeth. 74. Lewis' formula. 
75. Experimental data. 76. Teeth of bevel gears. 77. 
Rim and arms. 78. Sprocket wheels and chains, 79. 
Silent chains. 80. Cranks and levers. 

XI. Fly-Wheels 166 

81. In general. 82. Safe speed for wheels. 83. Experiments 
on fly wheels. 84. Wooden pulleys. 85. Rims of cast-iron 
gears. 86. Rotating discs. 87. Plain discs. 88. Conical 
discs. 89. Discs with logarithmic profile. 90. Bursting 
speeds. 

XII. Transmission by Belts and Ropes 184 

91. Friction of belting. 92. Strength of belting. 93. 
Taylor's experiments. 94. Rules for width of belts. 95. 
Speed of belting. 96. Manila rope transmission. 97. 
Strength of Manila ropes. 98. Wire rope transmission. 



TABLES. 



i. 

ii. 

in. 

Ilia. 

IV. 

V. 

VI. 

VII. 

VIII. 

IX. 

X. 

XI. 

XII. 

XIII. 

XIV. 

XV. 

XVI. 

XVII. 

XVIII. 

XIX. 

XX. 

XXI. 

XXII. 

XXIII. 

XXIV. 

XXV. 

XXVI. 

XXVII. 

XXVIII. 

XXIX. 

XXX. 

XXXI. 

XXXII. 

XXXIII. 

XXXIV. 



Strength of Wrought Metals 6 

Strength of Cast Metals. 7 

Values of Q in Column Formula 10 

Values of S and K in Column Formula 10 

Constants of Cross-Section 11 

Formulas for Loaded Beams 12 

Sizes of Iron and Steel Pipe 31 

Sizes of Extra Strong Pipe 33 

Sizes of Double Extra Strong Pipe 34 

Sizes of Iron and Steel Boiler Tubes 36 

Strength of Standard Screwed Pipe Fittings 41 

Bursting Strength of Cast Iron Cylinders 45 

Strength of Reinforced Cylinders 47 

Strength of Cast Iron Plates 52 

Strength of Iron or Steel Bolts 54 

Dimensions of Machine Screws 57 



Dimensions of Riveted Lap Joints 

Dimensions of Riveted Butt Joints 

Strength and Stiffness of Helical Springs . 
Friction of Piston Rod Packings 



.. 65 
.. 65 
.. 77 
.. 93 

" " " 94 

94 

Friction of Journal Bearing 108 

Friction of Roller anitPlajn Bearings 126 

127 

Diameters of Shafting 132 

Proportions of Gear Teeth 148 

Sizes of Test Fly-Wheels 172 

Flanges and Bolts of Test Fly-Wheels 172 

Failure of Flanged Joints 173 

Sizes of Linked Joints 173 

Failure of Linked Joints 174 

Bursting Speeds of Rotating Discs 182 

Horse Power of Manila Rope 192 

Horse Power of Wire Rope ... 194 

vii 



MACHINE DESIGN. 



CHAPTER I. 

UNITS AND TABLES. 

i. Units. In this book the following units will be 
used unless otherwise stated. 

Dimensions in inches. 

Forces in pounds. 

Stresses in pounds per square inch. 

Velocities in feet per second. 

Work and energy in foot pounds. 

Moments in pounds inches. 

Speeds of rotation in revolutions per minute. 

The word stress will be used to denote the resistance 
of material to distortion per unit of sectional area. 
The word deformation will be used to denote the dis- 
tortion of a piece per unit of length. The word set will 
be used to denote total permanent distortion of a piece. 

In making calculations the use of the slide-rale and 
of four-place logarithms is recommended ; accuracy is 
expected only to three significant figures. 

2. Abbreviations. The following abbreviations are 
among those recommended by a committee of the 
American Society of Mechanical Engineers in Decem- 
ber, 1904, and will be used throughout the book, 



MACHINE DESIGN. 



NAME. 

Inches 

Feet . 

Yards 

Miles . 

Pounds 

Tons 

Gallons 

Seconds 

Minutes 

Hours 

Linear 

Square 

Cubic 

Per 

Fahrenheit 

Percentage 

Brake horse power 

Electric horse power 

Indicated horse power 

British thermal units 

Diameter 



ABBREVIATION. 

. in. 
. ft. 
. yd. 

spell out. 
, lb. 
, spell out. 

gal. 

sec. 

min. 
, hr. 
, lin. 

sq. 

cu. 

spell out. 

fahr. 

% or per cent. 

b.h.p. 

e.h.p. 

i.h.p. 

B.t.u. 

Diam. 



3. Materials. The principal materials used in 
machine construction are given in the following tables 
with the physical characteristics of each. 

By wrought iron is meant commercially pure iron 
which has been made from molten pig-iron by the 
puddling process and then squeezed and rolled, thus 
developing the fiber. This iron has been largely sup- 
planted by soft steel. 

Ordinary wrought iron contains from 0.1% to 0.3% 
of carbon. Soft steel may contain no more than this, 
but is different in structure. The particles of iron in 
the puddling process are more or less enveloped in 
slag or earthy matter and as the bloom is squeezed and 



MATERIALS. 3 

rolled the particles become fibers separated from each 
other by a thin sheath or covering of slag, and it is 
this that gives such iron its characteristic structure. 
The principal impurities in the iron are phosphorus 
from the ore and sulphur from the fuel. 

In making steel, on the other hand, the molten 
iron has had the silicon and carbon removed by a hot 
blast, either passing through the liquid as in the Bes- 
semer converter, or over its surface as in the open- 
hearth furnace. A suitable quantity of carbon and 
manganese has then been added and the metal poured 
into ingot molds. If the steel is then reheated and 
passed through a series of rolls, structural steel and 
rods or rails result. 

Bessemer steel contains from 0.1% to 0.6% of carbon 
and has a fine granular structure. This material has 
been much used for rails. 

Open hearth steel differs from Bessemer but little in 
its chemical composition but is usually more reliable in 
quality on account of the more deliberate nature of 
the process of manufacture. It is generally used for 
boiler plate and for steel castings. Two grades of 
boiler plate are commonly known as marine steel and 
flange steel, the latter being of the better quality. 

Steel castings are poured directly from the open 
hearth furnace and allowed to cool without any draw- 
ing or rolling. They are coarser and more crystalline 
than the rolled steel. 

Crucible steel usually contains from one to one and 
a half per cent of carbon, is relatively high priced and 
only used for cutting tools. It is made by melting 
steel in an air-tight crucible with the proper additions 
of carbon and manganese. 

Cast iron is made directly from the pig by remelt- 



4 MACHINE DESIGN. 

ing and casting, is granular in texture and contains 
from two to five per cent, of carbon. A portion of the 
carbon is chemically combined with the iron while the 
remainder exists in the form of graphite. The harder 
and whiter the iron the more carbon is found chemi- 
cally combined. Silicon is an important element in 
cast iron and influences the rate of cooling. The more 
slowly iron cools after melting the more graphite forms 
and the softer the iron. 

Two per cent of silicon gives a soft gray iron of a 
high tensile strength. 

Machinery iron contains usually from one and one 
half to two per cent of silicon. 

Malleable iron is cast iron annealed and partially de- 
carbonized by being heated in an annealing oven in con- 
tact with some oxidizing material such as haematite ore. 
This process makes the iron tougher and less brittle. 

All castings including those made from alloys are 
somewhat unreliable on account of hidden flaws and of 
the strains developed by shrinkage while cooling. 

The so-called high-speed or air -hardening tool steels 
are alloys of steel with various substances such as 
chromium (chrome steel), tungsten (Mushet steel), 
molybdenum, etc., etc. 

They are characterized by extreme hardness at 
comparatively high temperatures. Their other physi- 
cal characteristics are not of particular interest. 

The addition of nickel to steel increases its ultimate 
strength and also raises its elastic limit. The tensile 
strength is sometimes as high as 200, 000 lb. per sq. in. aud 
the steel is also tough and well adapted to resist shocks. 

The bronzes are alloys of copper and tin, copper and 
zinc, or of all three. The copper- tin alloys usually 
contain 85 or 90 per cent of copper and are expensive. 



MATERIALS. 5 

The copper-zinc alloys, or brasses as they are sometimes 
called, should have from 60 to 70 per cent of copper 
for maximum strength and ductility. 

Bronzes high in tin and low in copper are weak, but 
have considerable ductility and make good metals for 
bearings. Tin 80, copper 10 and antimony 10 is Babbitt 
metal, so much used to line journal bearings, the 
antimony increasing the hardness. 

The late Dr. Thurston's experiments on the copper- 
tin-zinc alloys showed a maximum strength for copper 
55, zinc 13 and tin 2 per cent. The tensile strength of 
this mixture was nearly 70,000 lb. per sq. in. 

Phosphor bronze is a copper alloy with a small 
amount of phosphorus added to prevent oxidation of 
the copper and thereby strengthen the alloy. 

Manganese bronze is an alloy of copper and man- 
ganese, usually containing iron and sometimes tin. A 
bronze containing about 81 per cent copper, 11 per cent 
manganese and a little iron, has much the same physical 
characteristics as soft steel and resists corrosion much 
better. 

The constants for strength and elasticity given in 
the tables are only fair average values, and should be 
determined for any special material by direct experi- 
ment when it is practicable. Many of the constants are 
not given in the table on account of the lack of reliable 
data for their determination. 

The strength of steel, either rolled or cast, depends 
so much upon the percentages of carbon, phosphorus 
and manganese, that any general figures are liable to 
be misleading. Structural steel usually has a tensile 
strength of about 65,000 lb. per sq. in., while boiler 
plate usually has less carbon, a low tensile strength 
and good ductility. 



MACHINE DESIGN. 



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MACHINE DESIGN. 



4. Notation. 

Arc of contact 

Area of section 

Breadth of section 

Coefficient of friction 

Deflection of beam 

Depth of section 

Diameter of circular section 

Distance of neutral axis from outer fiber 

Elasticity, modulus of, 

in tension and compression 

in shearing and torsion 
Heaviness, weight per cu. ft. 
Length of any member 
Load or dead weight 
Moment, in bending 
in twisting 
Moment of inertia 

rectangular 

polar 
Pitch of teeth, rivets, etc. 
Eadius of gyration 

Section modulus, bending 

twisting 

Stress per unit of area 
Velocity 



-0 radians. 
-A sq. in. 
-b in. 

■/ 

=A in. 

■h in. 
-d in. 
-y in. 

E 
G 

-■w 
I in. 
Wlb. 
If lb. -in. 
= Tlb.-in. 

I 
J 

p in. 
■r in. 
J 

y 
.j 

y 

-s 

--v ft per sec. 



5. Formulas. 



Simple Stress. 



Tension, compression or shear, S — 



W 



(1) 



NOTATION. 

Bending under Transverse Load. 

or 
General equation, M = — . 

CfI7,2 

Rectangular section, M=- 



6 



Rectangular section, bh 2 = , 

SV7 3 
Circular section, M =^^ 



Circular section, d = r 



(2) . 
(3) 

m 

(5) 
(6) 



Torsion or Twisting. 



General equation, T=— (7) 

Circular section, ^ = f? ( 8 ) 

o. J. 

Circular section, d = lj \ . . . .(9) 

Hollow circular section, T ' =-~ d '~^ . .(10) 

Other values of - and - may be taken from Table 4. 
y V 

Combined Bending and Twisting. 

Calculate shaft for a twisting moment. 

T l =M + VM J TT~ 2 (11) 

Column subject to Bending. 

Use Rankine's formula, —t= w (12) 

The values of r 2 may be taken from Table IV. The 
subjoined table gives the average values of g, while $ 
is the compressive strength of the material. 



10 



machine design. 



TABLE III. 

Values of q in Formula 12. 



Material. 



Timber 

Cast Iron 

Wrought Iron 
Steel 



Both 
ends 
fixed. 


Fixed 

and 

round. 


Both 

ends 

round. 


1 


1.78 
3000 
1.78 
5000 
1.78 
36000 
1.78 
25000 


4 
3000 

4 
5000 

4 
36000 

4 
25000 


3000 

1 
5000 

1 
36000 

1 
25000 



Fixed 
and 
free. 



16 
3000 

16 
5000 

16 
36000 

16 
25000 



Carnegie's hand-book gives $ = 50000 for medium 



and 



18007 



for the 



steel columns and g=3 6-ioo, nk 
three first columns in above table. 

In this formula, as in all such, the values of the 
constant should be determined for the material used by 
direct experiment if possible. 

Or use straight line formula, -j— S — k- . . .(12a) 

TABLE Ilia. 

Values of S and k in Formula (12a). 

(Merriman's Mechanics of Materials.) 



Kind of Column. 


S 


k 


Limit - 
r 


Wrought Iron : 

Flat ends 


42000 
42000 
42000 

52500 
52500 
52500 

80000 
80000 
80000 

5400 


128 
157 
203 

179 
220 

284 

438 
537 
693 

28 


218 


Hinged ends 


178 


Round ends 

Mild Steel : 

Flat ends 


138 
195 


Hinged ends 


159 


Round ends.. 


123 


Cast Iron : 

Flat ends. 


122 


Hinged ends 


99 


Round ends. 


77 


Oak : 

Flat ends 


128 



FORMULAS. 



11 



Carnegie's hand-book gives allowable stress for 
structural columns of mild steel as 12000 for lengths 

less than 90 radii, and as 17100 — 57- for longer 

columns. 
This allows a factor of safety of about four. 

TABLE IV. 

Constants of Cross-Section. 



Form of 

Section and 

Area A. 


Square of 
Radius of 
Gyration 

r 2 


Moment 

of 
Inertia 

I=Ar* 


Section 

Modulus 

I 

y 


Polar 

Moment 

of Inertia 

J 


Torsion 

Modulus 

J 

y 


Rectangle 


/i 2 
12 


bh 3 
12 


bh* 
6 


bh 3 +b 3 h 


bh 3 +b 3 h 


bh 


12 


6 V b 2 +7i 2 


Square 
d 2 


d 2 
12 


d 4 
12 


d 3 
6 


d 4 
6 


d 3 
4.24 


Hollow- 
Rectangle 


bh*-b,h\ 


bh 3 -bji 3 x 


bh*—Vh\ 






or 7-beam 
bh—b 1 h 1 


Mlbh-bJiJ 


12 


Qh 




Circle 

l a 2 


d 2 
16 


7Td 4 

~BT 


d 3 
10.2 


7Td 4 

32 


d 3 
5.1 


Hollow 
Circle 


<P+d\ 
16 


7T(d 4 -d\) 


d*-d\ 
10. 2d 


7T(d 4 -d\) 


d*-d\ 


j(d*-d\) 


64 


32 


5, Id 


Ellipse 


a 2 
16 


rrba 3 
64 


6a 2 

10.2 


ir(ba 3 + ab 3 ) 


&a 3 -+-a& 3 


- A ab 

4 


64 


10.2a 



Values of I and J for more complicated sections can be worked 
out from those in table. 



12 



MACHINE DESIGN. 

TABLE V. 

Formulas for Loaded Beams. 



Beams of Uniform Cross-section. 


Maximum. 
Moment. 

M 


Maximum. 

Deflection. 

A 


Cantilever, load at end 


Wl 

Wl 
2 

Wl 

4 

Wl 

8 

dWl 
16 

Wl 

8 

Wl 

8 

Wl 
12 

Wl 

2 


WP 


Cantilever, uniform load 


3EI 
WP 


Simple beam, load at middle 


SEI 
WP 


Simple beam, uniform load 


4&EI 
§WP 


Beam fixed at one end, supported at other, 
load at middle 


384#J 

.0182TFZ 3 
EI 


Beam fixed at one end, supported at other, 
uniform load 


.0054 WT 
EI 


Beam fixed at both ends, load at middle. . 

Beam fixed at both ends, uniform load. . . . 

Beam fixed at both ends, load at one end, 
(pulley arm) 


WP 
192EI 

WP 
3UEI 

WP 
V2EI 



6. Profiles of Uniform Strength. In a bracket or 
beam of uniform cross-section the stress on the outer 
row of fibers increases as the bending moment increases 
and the piece is most liable to break at the point where 
the moment is a maximum. This difficulty can be 
remedied by varying the cross-section in such a way 
as to keep the fiber stress constant along the top or 
bottom of the piece. The following table shows the 
shapes to be used under different conditions. 



FACTORS OF SAFETY. 



13 



Type. 


Load. 


Plan. 


Elevation. 


Cantilever. . 


Center 


Rectangle . . 


Parabola, axis horizontal. 


Cantilever. . 


Uniform . . . 


Rectangle . . 


Triangle. 


Simp. Beam 


Center 


Rectangle . . 


Two parabolas intersecting 
under load. 


Simp. Beam 


Uniform . . . 


Rectangle . . 


Ellipse, major axis horizontal. 



The material is best economized by maintaining a 
constant breadth and varying the depth as indicated. 

This method of design is applicable to cast pieces 
rather than to those that are forged or cut. 

The maximum deflection of cantilevers and beams 
having a profile of uniform strength is greater than 
when the cross-section is uniform, fifty per cent, 
greater if the breadth varies, and one hundred per cent 
greater if the depth varies. 

7. Factors of Safety. A factor of safety is the ratio 
of the ultimate strength of any member to the ordinary 
working load which will come upon it. This factor is 
intended to allow for : (a) Overloading either inten- 
tional or accidental, (b) Sudden blows or shocks. 

(c) Gradual fatigue or deterioration of material. 

(d) Flaws or imperfections in the material. 

To a certain extent the term " factor of ignorance" 
is justifiable since allowance is made for the unknown. 
Certain fixed laws may guide one, however, in making 
the selection of a factor. It is a well-known fact that 
loads in excess of the elastic limit are liable to cause 
failure in time. Therefore, when the elastic limit of 
the material can be determined, it should be used as a 
basis rather than to use the ultimate strength. 



11 



MACHINE DESIGN. 



Furthermore, suddenly applied loads will cause 
about double the stress due to dead loads. These two 
considerations point to four as the least factor that 
should be used when the ultimate strength is taken as 
a basis. Pieces subject to stress alternately in opposite 
directions should have large factors of safety. 

The following table shows the factors used in good 
practice under various conditions : 



Structural steel in buildings . < 

" " " bridges 
Steel in machine construction 


4 
5 
6 


" " engine " 
Steel plate in boilers 
Cast iron in machines 


10 

5 

. 6 to 15 



Castings of bronze or steel should have larger factors 
than rolled or forged metal on account of the possibility 
of flaws. 

Cast iron should not be used in pieces subject to 
tension or bending if there is a liability of shocks or 
blows coming on the piece. 



CHAPTER IT. 



FRAME DESIGN. 



8. General Principles of Design. The working or 
moving parts should be designed first and the frame 
adapted to them. 

The moving parts can be first arranged to give the 
motions and velocities desired, special attention being 
paid to compactness and to the convenience of the 
operator. 

Novel and complicated mechanisms should be 
avoided and the more simple and well tried devices 
used. 

Any device which is new should be first tried in a 
working model before being introduced in the design. 

The dimensions of the working parts for strength 
and stiffness must next be determined and the design 
for the frame completed. This may involve some 
modification of the moving parts. 

In designing any part of the machine, the metal 
must be put in the line of stress and bending avoided 
as far as possible. 

Straight lines should be used for the outlines of 
pieces exposed to tension or compression, circular 
cross sections for all parts in torsion, and profile of 
uniform fiber stress for pieces subjected to bending 
action. 

Superfluous metal must be avoided and this excludes 
all ornamentation as such. There should be a good 

15 



16 MACHINE DESIGN. 

practical reason for every pound of metal in the 
machine. 

An excess of metal is sometimes needed to give 
inertia and solidity and prevent vibration, as in the 
frames of machines having reciprocating parts, like 
engines, planers, slotting machines, etc. 

Mr. Oberlin Smith has characterized this as the 
" anvil " style of design in contradistinction to the 
1 'fiddle" style. 

In one the designer relies on the mass of the metal, 
in the other on the distribution of the metal, to resist 
the applied forces. 

A comparison of the massive Tangye bed of some 
large high-speed engines with the comparatively slight 
girder frame used in most Corliss engines, will em- 
phasize this difference. 

It may be sometimes necessary to waste metal in 
order to save labor in finishing, and in general the aim 
should be to economize labor rather than stock. 

The designers should be familiar with all the shop 
processes as well as the principles of strength and 
stability. The usual tendency in design, especially of 
cast iron work, is towards unnecessary weight. 

All corners should be rounded for the comfort and 
convenience of the operator, no cracks or sharp inter- 
nal angles left where dirt and grease may accumulate, 
and in general special attention should be paid to so 
designing the machine that it may be safely and con- 
veniently operated, that it may be easily kept clean, 
and that oil holes are readily accessible. The ap- 
pearance of a machine in use is a key to its working 
condition. 

Polished metal should be avoided on account of its 
tendency to rust, and neither varnish nor bright colors 




Fig. 1. Old Planing Machine. An Example of Elaborate 
Ornamentation. 



GENERAL PRINCIPLES OF DESIGN. 17 

tolerated. The paint should be of some neutral tint 
and have a dead finish so as not to show scratches or 
dirt. 

Beauty is an element of machine design, but it can 
only be attained by legitimate means which are appro- 
propriate to the material and the surroundings. 

Beauty is a natural result of correct mechanical 
construction but should never be made the object of 
design. 

Harmony of design may be secured by adopting one 
type of cross-section and adhering to it throughout, 
never combining cored or box sections with ribbed 
sections. In cast pieces the thickness of metal should 
be uniform to avoid cooling strains, and for the same 
reason sharp corners should be absent. The lines of 
crystallization in castings are normal to the cooled 
surface and where two flat pieces come together at 
right angles, the interference of the two sets of crys- 
tals forms a plane of weakness at the corner. This is 
best obviated by joining the two planes with a bend or 
sweep. 

Eounding the external corner and filleting the in- 
ternal one is usually sufficient. Where two parts come 
together in such a way as to cause an increase of 
thickness of the metal there are apt to be " blow holes " 
or " hot spots" at the junction due to the uneven 
cooling. 

" Strengthening " flanges when of improper propor- 
tions or in the wrong location are frequently a source 
of weakness rather than strength. A cast rib or flange 
on the tension side of a plate exposed to bending, will 
sometimes cause rupture by cracking on the outer edge. 
When apertures are cut in a frame either for core- 
prints or for lightness, the hole or aperture should be 



18 MACHINE DESIGN. 

the symmetrical figure, and not the metal that sur- 
rounds it, to make the design pleasing to the eye. 

The design should be in harmony with the material 
used and not imitation. For example, to imitate 
structural work either of wood or iron in a cast-iron 
frame is silly and meaningless. 

Machine design has been a process of evolution. 
The earlier types of machines were built before the 
general introduction of cast-iron frames and had 
frames made of wood or stone, paneled, carved and 
decorated as in cabinet or architectural designs. 

When cast iron frames and supports were first 
introduced they were made to imitate wood and stone 
construction, so that in the earlier forms we find panels, 
moldings, gothic traceries and elaborate decorations of 
vines, fruit and flowers, the whole covered with con- 
trasting colors of paint and varnished as carefully as a 
piece of furniture for the drawing-room. Eelics of 
this transition period in machine architecture may be 
seen in almost every shop. One man has gone down 
to posterity as actually advertising an upright drill 
designed in pure Tuscan. 

9. Machine Supports. The fewer the number of 
supports the better. Heavy frames, as of large en- 
gines, lathes, planers, etc., are best made so as to rest 
directly on a masonry foundation. Short frames as 
those of shapers, screw machines and milling machines, 
should have one support of the cabinet form. The use 
of a cabinet at one end and legs at the other is offensive 
to the eye, being inharmonious. If two cabinets are 
used provision should be made for a cradle or pivot at 
one end to prevent twisting of the frame by an uneven 
foundation. The use of intermediate supports is 



GENERAL PRINCIPLES OF DESIGN. 19 

always to be condemned, as it tends to make the frame 
conform to the inequalities of the floor or foundation 
on what has been aptly termed the ' ' caterpillar prin- 
ciple." 

A distinction must be made between cabinets or 
supports which are broad at the base and intended to 
be fastened to the foundation, and legs similar to those 
of a table or chair. The latter are intended to simply 
rest on the floor, should be firmly fastened to the 
machine and should be larger at the upper end where 
the greatest bending moment will come. 

The use of legs instead of cabinets is an assumption 
that the frame is stiff enough to withstand all stresses 
that come upon it, unaided by the foundation, and if 
that is the case intermediate supports are unne- 
cessary. 

Whether legs or cabinets are best adapted to a cer- 
tain machine the designer must determine for himself. 

Where two supports or pairs of legs are necessary 
under a frame, it is best to have them set a certain 
distance from the- ends, and make the overhanging 
part of the frame of a parabolic form, as this divides 
up the bending moment and allows less deflection at 
the center. Trussing a long cast-iron frame with iron 
or steel rods is objectionable on account of the differ- 
ence in expansion of the two metals and the liability 
of the tension nuts being tampered with by work- 
men. 

The sprawling double curved leg which originated 
in the time of Louis XIV and which has served in turn 
for chairs, pianos, stoves and finally for engine lathes 
is wrong both from a practical and aesthetic stand- 
point. It is incorrect in principle and is therefore 
ugly. 



20 MACHINE DESIGN. 

Exercise. 

1. — Apply the foregoing principles in making a written 
criticism of some engine or machine frame and its supports. 

(a) Girder frame of engine. 

(b) Tangye bed of air compresser. 

(c) Bed, uprights and supports of iron planing machine. 

(d) Bed and supports of engine lathe. 

(e) Cabinet of shaping or milling machine. 

(f) Frame of upright drill. 

io. Machine Frames. For general principles of 
frame design the reader is referred to Chapter 2. Cast 
iron is the material most used but steel castings are 
now becoming common in situations where the stresses 
are unusually great, as in the frames of presses, shears 
and rolls for shaping steel. 

Cored vs. Rib Sections. Formerly the flanged or 
rib section was used almost exclusively, as but a few 
castings were made from each pattern and the cost of 
the latter was a considerable item. Of late years the 
use of hollow sections has become more common ; the 
patterns are more durable and more easily molded 
than those having many projections and the frames 
when finished are more pleasing in appearance. 

The first cost of a pattern for hollow work, including 
the cost of the core-box, is sometimes considerably 
more but the pattern is less likely to change its shape 
and in .these days of many castings from one pattern, 
this latter point is of more importance. Finally it 
may be said that hollow sections are usually stronger 
for the same weight of metal than any that can be 
shaped from webs and flanges. 



MACHINE FRAMES. 



21- 




Fig. 3. 



Resistance to Bending. Most machine frames are 
exposed to bending in one or two 
directions. If the section is to be 
ribbed it should be of the form 
shown in Fig. 3. The metal 
being of nearly uniform thickness 
and the flange which is in tension 
having an area three or four times 
that of the compression flange. 
In- a steel casting these may be 
more nearly equal. The hollow 
section may be of the shape shown in Fig. 4, a hollow 
rectangle with the tension side re-enforced and slightly 
thicker than the other three 
sides. The re-enforcing flanges 
at A and B may often be utilized 
for the attaching of other mem- 
bers to the frame as in shapers 
or drill presses. The box section 
has one great advantage over 
the I section in that its 
moment of resistance to side 
bending or to twisting is usually much greater. The 
double I or the U section is common where it is 
necessary to have two parallel 
ways for sliding pieces as in lathes 
and planers. As is shown in Fig. 
5 the two Is are usually connected 
at intervals by cross girts. 

Besides making the cross-section 
of the most economical form, it is 
often desirable to have such a 
longitudinal profile as shall give a 
uniform fiber stress from end to 




Fig. 4. 




22 MACHINE DESIGN. 

end. This necessitates a parabolic or elliptic outline 
of which the best instance is the housing or upright of 
a modern iron planer. 

A series of experiments made in 1902 under the 
direction of the author, on the modulus of rupture of 
cast iron beams of the same weight but different cross- 
sections gave interesting results. Beginning with the 
solid circular section, which failed under a transverse 
load of 7,500 lb., square, rectangular, hollow and 
I shaped sections were tested until a maximum was 
reached in the I section with heavy tension flange 
which broke under a load of 38,000 lb. Channel and 
T shaped sections such as are appropriate for fly-wheel 
rims were also tested with the ribs in tension and in 
compression. 

The strength of such sections was- found to be from 
two to three times as great when the ribs were in com- 
pression as when they were in tension. 

Resistance to Twisting. The 
hollow circular section is the ideal 
form for all frames or machine 
members which are subjected to 
torsion. If subjected also to 
bending the section may be made 
elliptical or, as is more common, 
thickened on two sides by making 
the core oval. See Fig. 6. As 
Fig. 6. has already been pointed out the 

box sections are in general better 
adapted to resist twisting than the ribbed or I sec- 
tions. 

Frames of Machine Tools. The beds of lathes are 
subjected to bending on account of their own weight 
and that of the saddle and on account of the downward 




MACHINE FRAMES. 



23 



pressure on the tool when work is being turned. They 
are usually subjected to torsion on account of the un- 
even pressure of the supports. The box section is then 
the best ; the double / commonly used is very weak 
against twisting. The same principle would apply in 
designing the beds of planers but the usual method of 
driving the table by means of a gear and rack prevents 
the use of the box section. The uprights of planers 
and the cross rail are subjected to severe bending 
moments and should have profiles of uniform strength. 
The uprights are also subject to side bending when the 
tool is taking a heavy side cut near the top. To pro- 
vide for this the uprights may be of a box section or 
may be reinforced by outside ribs. 

The upright of a drill press or vertical shaper is 
exposed to a constant bending moment equal to the 
upward pressure on the 
cutter multiplied by the 
distance from center of 
cutter to center of up- 
right. It should then be 
of constant cross-section 
from the bottom to the 
top of the straight part. 
The curved or goose- 
necked portion should 
then taper gradually. 

The frame of a shear press. or punch is usually of 
the G shape in profile with the inner fibers in tension 
and the outer in compression. The cross-section should 
be as in Fig. 3 or Fig. 4, preferably the latter, and 
should be graduated to the magnitude of the bending 
moment at each point. (See Fig. 7.) 




Fig. 7. 



24 MACHINE DESIGN. 

Exercises. 

1. Discuss the stresses and the arrangement of material in 
the girder frame of a Corliss engine. 

2. Ditto in the G frame of a band saw. 

Problem. 

Design a G frame similar to that shown in Fig. 7, for a shear 
press capable of shearing a bar of mild steel 1| by 1£ inches 
and having a gap four inches high and twenty-six inches deep. 



CHAPTER III. 



CYLINDER AND PIPES. 




Fig. 8. 



ii. Thin Shells. Let Fig. 8 represent a section of 
a thin shell, like a boiler 
shell, exposed to an inter- 
nal pressure of p pounds 
per sq. inch. Then, if we 
consider any diameter AB, 
the total upward pressure 
on upper half of the shell 
will balance the total down- 
ward pressure on the lower 
half and tend to separate 
the shell at A and B by- 
tension. 

Let d— diameter of shell in inches. 

r= radius of shell in inches. 
1= length of shell in inches. 
i= thickness of shell in inches. 
S= tensile strength of material. 

Draw the radial line CP to represent the pressure 
on the element P of the surface. 

Area of element at P=lrdO. 

Total pressure on element —plrdO. 

Vertical pressure on element =plr sin BdO. 

Total vertical pressure on APB= f plr sin 0d0=2plr. 

25 



26 MACHINE DESIGN. 

The area to resist tension at A and B=2tl and its 
total strength =2tlS. 

Equating the pressure and the resistance 
MS=2plr 

<-S -fi (18) 

The total pressure on the end of a closed cylindrical 
shell =7rr 2 p and the resistance of the circular ring of 
metal which resists this pressure =2irrtS. 

Equating : ZirrSt=Trr 2 p 

Therefore a shell is twice as strong in this direction 
as in the other. Notice that this same formula would 
apply to spherical shells. 

In calculating the pressure due to a head of water 
equals h, the following formula is useful : 

p = 0Ateh. . . . (15) 

In this formula h is in feet and p in pounds per 
square inch. 

Problems. 

1. A cast-iron water pipe is 12 inches in internal diameter 
and the metal is .45 inches thick. What would be the factor 
of safety, with an internal pressure due to a head of water of 
250 feet ? 

2. What would be the stress caused by bending due to 
weight, if the pipe in Ex. 1 were full of water and 24 feet long, 
the ends being merely supported ? 

3. A standard lap-welded steam pipe, 8 inches in nominal 
diameter is 0.32 inches thick and is tested with an internal 
pressure of 500 pounds per sq. inch. What is the bursting 
pressure and what is the factor of safety above the test pres- 
sure, assuming £=40000 ? 



THICK SHELLS. 27 

12. Thick Shells. There are several formulas for 
thick cylinders and no one of them is entirely satis- 
factory. It is however generally admitted that the 
tensile stress caused by internal pressure in such a 
cylinder is greatest at the inner circumference and 
diminishes according to some law from there to the 
exterior of the shell. This law of variation is expressed 
differently in the different formulas. 

Barloitfs Formulas. Here the cylinder diameters 
are assumed to increase under the pressure, but in such 
a way that the volume of metal remains constant. 
Experiment has proved that in extreme cases this last 
assumption is incorrect. Within the limits of ordinary 
practice it is, however, approximately true. 

Let d x and.c? 2 be the interior and exterior diameters 

d —d 
in inches and let t= 2 ' ■ be the thickness of metal. 
A 

Let I be the length of cylinder in inches. 

Let Si and S 2 be the tensile stresses in lbs. per sq. 
inch at inner and outer circumferences. 

The volume of the ring of metal before the pressure 
is applied will be : 

V x =^{d>-d x >) 

and if the two diameters are assumed to increase the 
amounts x t and x 2 under pressure the final volume 
will be : 

Assuming the volume to remain the same : 
d* - d x 2 = (d 2 + x 2 f - (4 + Xif 



28 MACHINE DESIGN. 

Neglecting the squares of x x and x % this reduces to : 

U/iOCi &2QC2 

or the distortions are inversely as the diameters. 
The unit deformations will be proportional to 

% and §? 
d 2 a 

and the stresses >Si and S 2 will be in the same ratio : 

Si^xA^dl ( a ) 

02 0C 2 (X\ 0/\ 

or the stresses vary inversely as the squares of the 
diameters. Let S be the stress at any diameter d, 
then : 

S=^l=^L (where r is radius) 

and the total stress on an element of the area l.dr is : 

Integrating this expression between the limits ~ 
and ~ for r and multiplying by 2 we have : 

P=2 ^H> 2 ^OT (6) 

Equating this to the pressure which tends to produce 
rupture, pdl, where p is the internal unit pressure, 
there results : 

p-^L (16) 

2 St 
The formula (13) for thin shells gives p = —^-- 

By comparing this with formula (16) it will be seen 
that in designing thick shells the external diameter 
determines the working pressure or : 

P JM ( 16 «) 

C*2 



THICK SHELLS. 



29 



Lame's Formula. — In this discussion each particle 
of the metal is supposed to be subjected to radial com- 
pression and to tangential and longitudinal tension and 
to be in equilibrium under these stresses. 

Using the same notation 
as in previous formula : 
S - d 2 2 + d* /-.^ 

* i -d:-d? Pi ■ ■ K U) 

for the maximum stress 
at the interior, 

(18) 



and s ' = d7=^ p - 

for the stress at the outer 
surface. 

Fig. 9 illustrates the 
variation in S from inner 
to outer surface. 

Solving for d 2 in (17) we have 




Fig. 9. 



d a 



d / fl+P- . 



.(19) 



A discussion of Lame's formula may be found in 
most works on strength of materials. 

Problems. 

1. A hydraulic cylinder has an inner diameter of 8 inches, a 
thickness of four inches and an internal pressure of 1500 lbs. 
per sq. in. Determine the maximum stress on the metal by 
Barlow's and Lame's formulas. 

2. Design a cast-iron cylinder 6 inches internal diameter to 
carry a working pressure of 1200 lbs. per sq. in. with a factor 
of safety of 10. 

3. A cast-iron water pipe is 1 inch thick and 12 inches in- 
ternal diameter. Required head of water which it will carry 
with a factor of safety of 6. 



30 MACHINE DESIGN. 

13. Steel and Wrought Iron Pipe. Pipe for the 
transmission of steam, gas or water may be made of 
wrought iron or steel. Cast-iron is used for water 
mains to a certain extent, but its use for either steam 
or gas has been mostly abandoned. The weight of 
cast-iron pipe and its unreliability forbid its use for 
high pressure work. 

Wrought iron pipe up to and including one inch in 
diameter is usually butt-welded, and above that is lap- 
welded. Steel pipes may be either welded or may be 
drawn without any seam. Electric welding has been 
successfully applied to all kinds of steel tubing, both 
for transmitting fluids and for boiler tubes. 

The following tables are taken by permission from 
the catalogue of the Crane Company and show the stand- 
ard dimensions for steam pipe and for boiler tubes. 

Ordinary standard pipe is used for pressures not 
exceeding 100 lb. per sq. in., extra strong pipe for the 
pressures prevailing in steam plants where compound 
and triple expansion engines are used, while the 
double extra is employed in hydraulic work under the 
heavy pressures peculiar to that sort of transmission. 



STEEL AND WROUGHT IRON PIPE. 



31 





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d 
o 
ft 



STEEL AND WROUGHT IRON PIPE. 35 

Tests made by the Crane Company on ordinary com- 
mercial pipe such as is listed in Table VI showed the 
following pressures : 

8 in. diam. . . 2000 lb. per sq. in. 

10 " . . 2300 

12 " . . 1500 " " 

The pipe was not ruptured at these pressures. 



36 



MACHINE DESIGN. 



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STEEL AND WROUGHT IRON PIPE. 



37 



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38 MACHINE DESIGN. 

14. Strength of Boiler Tubes. When tubes are used 
in a so-called fire-tube boiler with the gas inside and the 
water outside, they are exposed to a collapsing pres- 
sure. 

The same is true of the furnace flues of internally 
fired boilers. Such a member is in unstable equilibrium 
and it is difficult to predict just when failure will 
occur. 

Experiments on small wrought-iron tubes have 
shown the collapsing pressure to be about 80 per cent. 
of the bursting pressure. With short tubes set in 
tube sheets the length would have considerable in- 
fluence on the strength, but ordinary boiler tubes 
collapsing at the middle of the length would not be 
influenced by the setting. 

The strength of such tubes is probably proportional 

to l-jj where t is the thickness and d the diameter. 

D. K. Clark gives for large iron flues the following 
formula : 

p= 2o y'. (20) 

where P is the collapsing pressure in lb. per sq. in. 
These flues had diameters varying from 30 in. to 50 in. 
and thickness of metal from f in. to T V in. 

Prof. R. T. Stewart has recently made some in- 
teresting experiments on the collapsing pressure of 
lap- welded steel tubes and reported the results to the 
American Society of Mechanical Engineers. (See 
Transactions, Vol. XXVII.) 

It will only be possible here to give some of the gen- 
eral conclusions, as stated by the author in his paper : 

1. The length of tube, between transverse joints 
tending to hold it to a circular form, has no practical 



STRENGTH OF BOILER TUBES. 39 

influence upon the collapsing pressure of a commercial 
lap- welded steel tube so long as this length is not less 
than about six diameters of tube. 

2. The formulae, as based upon the present research, 
for the collapsing pressure of modern lap- welded Bes- 
semer steel tubes, are as follows : 

P = 1000(l- J 1-1600 j 2 .) . .-.(A) 

' p = 86670-g- (B) 

Where P = collapsing pressure, pounds per sq. inch. 

d= outside diameter of tube in inches 

t= thickness of wall in inches 
Formula (A) is for values of P less than 581 pounds, 

or for values of -j less than 0.023, while formula (B) 

is for values greater than these. 

These formulae, while strictly correct for tubes that 
are 20 feet in length between transverse joints tending 
to hold them to a circular form, are, at the same time, 
substantially correct for all lengths greater than about 
six diameters. 

They have been tested for seven diameters, ranging 
from 3 to 10 inches, in all obtainable thicknesses of 
wall, and are known to be correct for this range. 

3. The apparent fiber stress under which the different 
tubes failed varied from about 7000 pounds for the 
relatively thinnest to 35000 pounds per square inch for 
the relatively thickest walls. 

Since the average yield point of the material was 
37000 and the tensile strength 58000 pounds per square 
inch, it would appear that the strength of a tube 
subjected to a collapsing fluid pressure is not dependent 



40 MACHINE DESIGN. 

alone upon either the elastic limit or ultimate strength 
of the material constituting it. 

15. Pipe Fittings. Steam pipe up to and including 
pipe two inches in diameter is usually equipped with 
screwed fittings, including ells, tees, couplings, valves, 
etc. 

Pipe of a larger size, if used for high pressures, 
should be put together with flanged fittings and bolts. 
One great advantage of the latter system is the fact 
that a section of pipe can easily be removed for repairs 
or alterations. 

' Small connections are usually made of cast-iron or 
malleable iron. While the latter are neater in ap- 
pearance they are more apt to stretch and cause leaky 
joints. The larger fittings are made of cast-iron or 
cast-steel. Such fittings can be obtained in various 
weights and thicknesses, to correspond to those grades 
of pipe listed in the tables. 

The designer should have at hand catalogues of pipe 
fittings from the various manufacturers, as these will 
give in detail the proportions of all the different con- 
nections. 

For pressures not exceeding 100 lbs. per sq. in. rubber 
and asbestos gaskets can be used between the flanges, 
but for higher pressures or for superheated steam 
corrugated metallic gaskets are necessary. 

In 1905 some very interesting experiments on the 
strength of standard screwed elbows and tees were 
made by Mr. S. M. Chandler, a graduate of the Case 
School, and published by him in " Power " for October, 
1905. 

The fittings were taken at random from the stock of 
the Pittsburg Valve and Fittings Co., and three of 



PIPE FITTINGS. 



41 



each size were tested to destruction by hydraulic pres- 
sure. 

The following table gives a summary of the results 
obtained. The values which are starred in the table 
were obtained from fittings which had purposely been 
cast with the core out of center so as to make one wall 
thinner than the other. These values are not included 
in the averages. 

TABLE X. 

BURSTING STRENGTH OF STANDARD SCREWED FITTINGS, 
PRESSURES IN POUNDS PER SQUARE INCH. 

SIZE. ELBOWS. AVERAGE. 




SIZE. 



TEES. 



AVERAGE. 



li 


3400 


3300 


3300 


3333 


H 


3400 


3200 


2800* 


3300 


2 


2500 


2800 


2500 


2600 


8* 


2400 


2100* 


2500 


2450 


3 


1400* 


1900 


1800 


1850 


3i 


1200* 


1500 


1800 


1650 


4 


1800 


2100 


1700 


1867 


U 


1100* 


1400 


1400 


1400 


5 


1700 


1300* 


1500 


1600 


6 


1400 


1500 


1100* 


1450 


7 


1400 


1400 


1500 


1433 


8 


1200* 


1400 


1300 


1350 


9 


1300 


1400 


1200 


1300 


10 


1100 


1300 


1200 


1200 


12 


1100 


1000 


1100 


1067 



* Made with eccentric core. 



42 MACHINE DESIGN. 

These tests show a large apparent factor of safety 
for any pressures to which screwed fittings are usually 
subjected. 

The failure of such fittings in practice must be at- 
tributed to faulty workmanship in erection, such as 
screwing too tight, lack of allowance for expansion 
and poor drainage. 

The average tensile strength of the cast-iron used in 
the above fittings was 20000 lbs. per sq. in. 

Problems. 

1. Determine the bursting pressure of a wrought iron steam 

pipe 6 inches nominal diameter. 

(a) If of standard dimensions. 

(b) If extra strong. 

(c) If double extra strong. 

2. Compare the above with the strength of standard screwed 

elbows and tees of the same size. * 

3. Determine the probable collapsing pressure of a charcoal 

iron boiler- tube of two inches nominal diameter. 

16. Steam Cylinders. Cylinders of steam engines 
can hardly be considered as coming under either of the 
preceeding heads. On the one hand the thickness of 
metal is not enough to insure rigidity as in hydraulic 
cylinders, and on the other the nature of the metal 
used, cast-iron, is not such as to warrant the assump- 
tion of flexibility, as in a thin shell. Most of the for- 
mulas used for this class of cylinder are empirical and 
founded on modern practice. 

Van Buren's formula for steam cylinders is : 

*tf=.0001_pd + .15i/3 (21) 



* See Whitham's " Steam Engine Design," p. 27. 



STEAM CYLINDERS. 



43 



A formula which the writer has developed is some- 
what similar to Van Buren's. 

Let s' = tangential stress due to internal pressure. 
Then by equation for thin shells 

6 2t 

Let s" be an additional tensile stress due to distor- 
tion of the circular section at any weak point. 

Then if we regard one-half of the circular section 
as a beam fixed at A and B (Fig. 11) and assume the 
maximum bending moment as at C some weak point, 
the tensile stress on the outer 
fibres at C due to the bending 

pd 2 
f 



will be proportional to 

by the laws of flexure, or 
„_cpd* 



f 



unknown 



where c is some 
constant. 

The total tensile stress at 
C will then be 




£=s' + s 



„_pd 



2t 



cpcV 



Solving for c 
Solving for t 



c= 



pd 2 



t_ 

2d- 



t 



pd \cpd 2 



+ 



p\V 
16S* 



.(a) 
(22) 



a form which reduces to that of equation (13) when 
c=0. 

An examination of several engine cylinders of 



44 MACHINE DESIGN. 

standard manufacture shows values of c ranging from 
.03 to .10, with an average value : 

c=.06. 

The formula proposed by Professor Barr, in his 
paper on* " Current Practice in Engine Proportions," 
as representing the average practice among builders of 
low speed engines is : 

£=.05 d+. 3 inch (23) 

In Kent's Mechanical Engineer's Pocket Book, the 
following formula is given as representing closely 
existing practice : 

£=.0004 dp + 0.3 inch (24) 

This corresponds to Barr's formula if we take p= 125 
pounds per square inch. 

Experiments f made at the Case School of Applied 
Science in 1896-97 throw some light on this subject. 
Cast iron cylinders similar tp those used on engines 
were tested to failure by water pressure. The cylinders 
varied in diameter from six to twelve inches and in 
thickness from one-half to three-quarters inches. 

Contrary to expectations most of the cylinders failed 
by tearing around a circumference just inside the 
flange. (See Fig. 12). 



* Transactions A. S. M. E., vol. xviii, p. 741. 
t Transactions A. S. M. E. vol. xix. 



STEAM CYLINDERS. 
TABLE XI. 



45 





Diam. 
d 


Pres- 
sure. 
P 


Thick- 
ness. 
t 


Line 

of 

Failure . 


Formulas Used. 


Strength 


No. 


13 
q _pd 
°~2t 


14 
s= pd 
U 


a 

C — 


of 
Test-bar. 


a 


12.16 


800 


.70 


Circum. 


0940 


3470 


.046 


18000 lbs. 


d 


12.45 


700 


.56 


Longi. 


7780 





.047 


24000 lbs. 


e 


9.12 


1325 


.61 


Circum. 


9900 


4950 


.048 


24000 lbs. 


f 


6.12 


2500 


.65 


Circum. 


11800 


5900 


.055 


24000 lbs. 


1 


9.58 


600 


.402 


Longi. 


7150 





.049 


24000 lbs. 


2 


9.375 


1050 


.573 


Circum. 


8590 


4300 


.055 


24000 lbs. 


3 


9.13 


975 


.596 


Circum. 


7470 


3740 


.072 


24000 lbs. 


4 


12.53 


700 


.571 


Longi. 


7680 





.048 


24000 lbs. 


5 


12.56 


875 


.531 


Circum. 


10350 


5180 


.028 


24000 lbs. 



Average of c=.05 

Table XI gives a summary of the results. 

Out of nine cylinders so tested, only three failed by 
splitting longitudinally. 

This appears to be due to two causes. In the first 
place, the flanges caused a bending moment at the 
junction with the shell due to the pull of the bolts. 
In the second place, the fact that the flanges were 
thicker than the shell caused a zone of weakness near 
the flange due to shrinkage in cooling, and the pres- 
ence of what founders call " a hot spot." 

The stresses figured from formula (14) in the cases 
where the failure was on a circumference, are from 
one-fifth to one-sixth the tensile strength of the test 
bar. 



46 MACHINE DESIGN. 

The strength of a chain is the strength of the 
weakest link, and when the tensile stress exceeded the 
strength of the metal near some blow hole or "hot 
spot," tearing began there and gradually extended 
around the circumference. 

Values of c as given by equation (a) have been cal- 
culated for each cylinder, and agree fairly well, the 
average value being c = .05. 

To the criticism that most of the cylinders did not 
fail by splitting, and that therefore formulas (a) and 
(22) are not applicable, the answer would be that the 
chances of failure in the two directions seem about 
equal, and consequently we may regard each cylinder 
as about to fail by splitting under the final pressure. 

If we substitute the average value of c=.05 and a 
safe value of 5=2000, formula (21) reduces to : 



,_ pd . d \ p 2 / 2r x 

1 8000 + 200\- p+ 1600 K } 

* Subsequent experiments made at the Case School 
in 1904 show the effect of stiffening the flanges by 
brackets. 

Four cylinders were tested, each being 10 inches 
internal diameter by 20 inches long and having a 
thickness of about f inches. The flanges were of the 
same thickness as the shell and were re-enforced by 
sixteen triangular brackets as shown in Fig. 13. 

The fractures were all longitudinal there being but 
little of the tearing around the shell which was. so 
marked a feature of the former experiments. This 
shows that the brackets served their purpose. 

Table XII gives the results of the tests and the 
calculated values of c. 



* Machinery, N. Y., Nov. 1905. 







j^i',. 


fim 




^Wkjg>- 






M 


1 f ^'.-^fc,. 


W."^T7^ 


-• '.''Mf'' / 



Fig. 12. Fractured Cylinder. 




Fig. 13. Fractured Cylinder. 



STEAM CYLINDERS. 
TABLE XII. 

BURSTING PRESSURE OF CAST-IRON CYLINDERS. 



47 



Internal 
Diameter. 


Average 
Thickness. 


Bursting 
Pressure. 


Value 

of c. 


s=1 4- 


10.125 


0.766 


1350 


.0213 


9040 


10.125 


0.740 


1400 


.0152 


10200 


10.125 


0.721 


1350 


.0126 


9735 


10.125 


0.720 


1200 


.0177 


9080 



Average value of c=.0167. 

Comparing the values in the above table with those 
in Table XI we find c to be only one-third as large. 

The tensile strength of the metal in the last four 
cylinders, as determined from test bars, was only 
14000 lbs. per sq. in. 

Comparison with the values of $due to direct tension 
as given by the formula 

pel 
2t 



S 



shows that in a cylinder of this type about one-third 
of the stress is " accidental" and due to lack of uni- 
formity in the conditions. In Table XI about two- 
thirds must be thus accounted for. 

Problems. 

1. Referring to Table XI, verify in at least three experiments 
the values of S and c as there given. Do the same in Table 
XII. 

2. The steam cylinder of a Baldwin locomotive is 22 ins. in 
diameter and 1.25 ins. thick. Assuming 125 lbs. gauge pres- 



48 MACHINE DESIGN. 

sure, find the value of c. Calculate thickness by Van Buren's 
and Barr's formulas. 

3. Determine proper thickness for cylinder of cast-iron, if 
the diameter is 38 inches and the steam pressure 100 lbs. by 
formulas 13, 21, 23, 24 and 25. 

4. The cylinder of a stationary engine has internal diameter 
=12 in. and thickness of shell =1 in. Find the value of c for 
p=120 lbs. per sq. in. 

17. Thickness of Flat Plates. An approximate 
formula for the thickness of flat cast-iron plates may 
be derived as follows : 

Let I— length of plate in inches. 

b= breadth of plate in inches. 

£= thickness of plate in inches. 

p— intensity of pressure in pounds. 

S = modulus of rupture lbs. per sq. in. 

A plate which is supported or fastened at all four 
edges is constrained so as to bend in two directions at 
right angles. Now if we suppose the plate to be 
represented by a piece of basket work with strips 
crossing each other at right angles we may consider 
one set of strips as resisting one species of bending 
and the other set as resisting the other bending. We 
may also consider each set of strips as carrying a 
fraction of the total load. The equation of condition 
is that each pair of strips must have a common deflec- 
tion at the crossing. 

Suppose the plate to be divided lengthwise into flat 
strips an inch wide I inches long, and suppose that a 
fraction p' of the whole pressure causes the bending of 
these strips. 



THICKNESS OF FLAT PLATES. 49 

Regarding the strips as beams with fixed ends and 
uniformly loaded : 

q SM__§Wl_p'V 
^ bh 2 V2bh 2 2f 

and the thickness necessary to resist bending is : 

<=4& ^ 

In a similar manner, if we suppose the plate to be 
divided into transverse strips an inch wide and b inches 
long, and suppose the remainder of the pressure p—p' 
equals p" to cause the bending in this direction, we 
shall have : 



4& <*> 



t = b 

But as all these strips form one and the same plate 
the ratio of p' to p" must be such that the deflection 
at the center of the plate may be the same on either 
supposition. The general formula for deflection in this 
case is 

wr 



A = 



SS±EI 



and J=to for each set of strips. Therefore the deflec- 

tion is proportional to ^-and ^» in the two cases. 

.-. p'l"=p"b" 
But p fJ rp" = p 

Solving in these equations for p' and p" 

P J4 +& 4 



50 MACHINE DESIGN. 

Substituting these values in (a) and (b) 



'- Wsscfor (26) 

*' w 4mfm (27) 

As l>b usually, equation (27) is the one to be used. 
If the plate is square l=b and 



*-f 



>fe ^ 



If the plate is merely supported at the edges then 
formulas (26) and (27) become : 
For rectangular plate : 

1 2 \ £(** + &*)' ' K } 

For square plate : 

<=-hl¥ w> 

A round plate may be treated as square, with 
side = diameter, without sensible error. 

The preceding formulas can only be regarded as 
approximate. G-rashof has investigated this subject 
and developed rational formulas but his work is too 
long and complicated for introduction here. His for- 
mulas for round plates are as follows : 

Bound plates : 
Supported at edges : 

4V¥ • • -^ 



t 



Fixed at edges : 



>fe • • ■<*> 



THICKNESS OF FLAT PLATES. 51 

where t and p are the same as before, d is the diameter 
in inches and S is the safe tensile strength of the 
material. 

Comparing these formulas with (28) and (30) for 
square plates, they are seen to be nearly identical if 
allowance is made for the difference in the value of S. 

Experiments made at the Case School of Applied 
Science in 1896-97 on rectangular cast iron plates with 
load concentrated at the center gave results as follows : 
Twelve rectangular plates planed on one side and each 
having an unsupported area of 10 by 15 inches were 
broken by the application of a circular steel plunger 
one inch in diameter at the geometrical center of each 
plate. The plates varied in thickness from one-half 
inch to one and one-eighth inches. Numbers 1 to 6 
were merely supported at the edges, while the remain- 
ing six were clamped rigidly at regular intervals 
around the edge. 

To determine the value of S, the modulus of rup- 
ture of the material, pieces were cut from the edge of 
the plates and tested by cross-breaking. The average 
value of S from seven experiments was found to be 
33000 lbs. per sq. in. 

In Table XIII are given the values obtained for the 
breaking load W under the different conditions. 

If we assume an empirical formula : 

W = k F+lf ( a ) 

and substitute given values of S, I and b we have 
nearly : 

W = 100kf (b) 

Substituting values of W and t from the Table XIII 
we have the values of k as given in the last column. 



52 



MACHINE DESIGN. 



If we average the values for the two classes of plates 
and substitute in (a) we get the following empirical 
formulas : 

For breaking load on plates supported at the edges 
and loaded at the center : 



JT=276 



St 2 
l 2 + V 



and for similar plates with edges fixed : 



W=U2 



St 2 

l 2 +b 2 



(31) 

(32) 



S in both formulas is the modulus of rupture. 

TABLE XIII. 

CAST IRON PLATES 10x15 INS. 





Thickness 


Breaking 


Constant. 


No. 


• 


Load. 






t 


W 


k 


1 


.562 


7500 


237 


2 


.641 


11840 


288 


3 


.745 


14800 


267 


4 


.828 


21900 


320 


5 


1.040 


31200 


289 


6 


1.120 


31800 


254 


7 


.481 


9800 


424 


8 


.646 


17650 


422 


9 


.769 


26400 


446 


10 


.881 


33400 


430 


11 


1.020 


47200 


454 


12 


1.123 


59600 


477 



Those plates which were merely supported at the 
edges broke in three or four straight lines radiating 
from the center. Those fixed at the edges broke in 
four or five radial lines meeting an irregular oval 
inscribed in the rectangle. Number 12 however failed 
by shearing, the circular plunger making a circular 
hole in the plate with several radial cracks. 



THICKNESS OF FLAT PLATES. 53 

Some tests were made in the spring of 1906 at the 
Case School laboratories by Messrs. Hill and Nadig on 
the strength of flat cast-iron plates under uniform 
hydraulic pressure. 

The plates tested were of soft gray iron, having a 
low tensile strength of about 12000 lbs. per square 
inch, and were of the following sizes : 
12 by 12 by f inches. 
12 by 12 by 1 inches. 
12 by 18 by 1.25 inches. 
12 by 18 by |f inches. 
These burst at the following pressures respectively : 

375 lbs. 675 lbs. 650 lbs. 450 lbs. 
The fractures started at the center of the plates and 
ran to the sides in irregular lines. The square plates 
were somewhat weaker than would have been expected 
from the formula and the rectangular plates somewhat 
stronger. 

Problems. 

1. Calculate the thickness of a steam- chest cover 8 X 12 
inches to sustain a pressure of 90 lbs. per sq. inch with a factor 
of safety =10. 

2. Calculate the thickness of a circular manhole cover of 
cast-iron 18 inches in diameter to sustain a pressure of 150 lbs. 
per sq. inch with a factor of safety =8, regarding the edges as 
merely supported. 

3. Determine the probable breaking load for a plate 18 by 
24 in. loaded at the center, (a) when edges are fixed, (b) 
When edges are supported. 

4. In experiments on steam cylinders, a head 12 inches in 
diameter and 1.18 inches thick failed under a pressure of 900 
lbs. per sq. in. Determine the value of S by formula (28). 



CHAPTEE IV. 



FASTENINGS. 



18. Bolts and Nuts. Tables of dimensions for U. S. 
standard bolt heads and nuts are to be found in most 
engineering hand-books and will not be repeated here. 

These proportions have not been generally adopted 
on account of the odd sizes of bar required. The 
standard screw-thread has been quite generally ac- 
cepted as superior to the old V-thread. 

Eoughly the diameter at root of thread is 0.83 of the 
outer diameter in this system, and the pitch in inches 
is given by the formula 

p=.24:\/d+ 625-.175. . . '. . .(33) 

where d= outer diameter. 

TABLE XIV. 

SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS. 



Diam. 
of 


Thr'ds 
per 
Inch. 

No. 


Diam. 

at 
Root of 
Thread. 
Inches. 


Area 

at 

Root of 

Thread. 

Sq. In. 


Safe Load in 
Tension. Lb. 


Safe Load in 
Shear. Lb. 


Bolt. 
Inch. 


5000 lb. 
per sq. in. 

135 


7500 lb. 
per sq. in. 


4000 lb. 
per sq. in. 


6000 lb. 
per sq. in. 


i 


20 


.185 


.0269 


202 


196 


294 


5 


18 


.240 


.0452 


226 


340 


306 


460 


1 


16 


.294 


.0679 


340 


510 


440 


660 


7 
T6 


14 


.344 


.0930 


465 


695 


600 


900 


i 


13 


. .400 


.1257 


628 


940 


785 


1175 



54 



BOLTS AND NUTS. 



55 



TABLE XIV (Continued). 
SAFE WORKING STRENGTH OF IRON OR STEEL BOLTS. 



Diam. 
of 


Thr'ds 

per 
Inch. 

No. 


Diam. 

at 
Root of 
Thread. 
Inches. 


Area 

at 

Root of 

Thread. 

Sq. In. 


Safe Load in 
Tension. Lb. 


Safe Load in 
Shear. Lt. 


Bolt. 
Inch. 


5000 lb. 
per sq. in. 


7500 lb. 
per sq. in. 


4000 lb, 
per sq. in. 


6000 lb. 
per sq. in. 


9 
T6 


12 


.454 


.162 


810 


1210 


990 


1485 


1 


11 


.507 


.202 


1010 


1510 


1230 


1845 


f 


10 


.620 


.302 


1510 


2260 


1770 


2650 


I 


9 


.731 


.420 


2100 


3150 


2400 


3600 




8 


.837 


.550 


2750 


4120 


3140 


4700 


H 


7 


.940 


.694 


3470 


5200 


3990 


6000 


■li 


7 


1.065 


.891 


4450 


6680 


4910 


7360 


If 


6 


1.160 


1.057 


5280 


7920 


5920 


7880 


1* 


6 


1.284 


1.295 


6475 


9710 


7070 


10600 


It 


5* 


1.389 


1.515 


7575 


11350 


8250 


.12375 


If 


5 


1.490 


1.744 


8720 


13100 


9630 


14400 


11 


5 


1.615 


2.049 


10250 


15400 


11000 


16500 


2 


4| 


1.712 


2.302 


11510 


17250 


12550 


18800 



The shearing load is calculated from the area of the 
body of the bolt. 

Bolts may be divided into three classes which are 
given in the order of their merit. 

1. Through bolts, having a head on one end and a 
nut on the other. 

2. Stud bolts, having a nut on one end and the other 
screwed into the casting. 

3. Tap bolts or screws having a head at one end and 
the other screwed into the casting. 



56 MACHINE DESIGN. 

The principal objection to the last two forms and 
especially to (3) is the liability of sticking or breaking 
off in the casting. 

Any irregularity in the bearing surfaces of head 
or nut where they come in contact with the casting, 
causes a bending action and consequent danger of 
rupture. 

This is best avoided by having a slight annular 
projection on the casting concentric with the bolt hole 
and finishing the flat surface by planing or counter- 
boring. 

Counter-boring without the projection is a rather 
slovenly way of overcoming the difficulty. 

When bolts or studs are* subjected to severe stress 
and vibration, it is well to turn down the body of the 
bolt to the diameter at root of thread, as the whole 
bolt will then stretch slightly under the load. 

A check nut is a thin nut screwed firmly against the 
main nut to prevent its working loose, and is commonly 
placed outside. 

As the whole load is liable to come on the outer nut, 
it would be more correct to put the main nut outside. 
(Prove this by figure.) 

After both nuts are firmly screwed down, the outer 
one should be held stationary and the inner one reversed 
against it with what force is deemed safe, that the 
greater reaction may be between the nuts. 

Numerous devices have been invented for the purpose 
of holding nuts from working loose under vibration 
but none of them are entirely satisfactory. 

Probably the best method for large nuts is to drive 
a pin or cotter entirely through nut and bolt. 

A flat plate, cut out to embrace the nut and fastened 
to the casting by a machine screw, is often used. 



MACHINE SCREWS. 



57 



of four shapes, the 



Vf7 



19. Machine Screws. A screw is distinguished from 
a bolt by having a slotted, round head instead of a 
square or hexagon head. 

The head may have any one 
round, fillister, oval fillister and 
fiat as shown in Fig. 14. A 
committee of the American 
Society of Mechanical Engineers 
has recently recommended cer- 
tain standards for machine 
screws.* The form of thread 
recommended is the U. S. Stand- 
ard or Sellers type with provision 
for clearance at top and bottom 
to insure bearing on the body of 
the thread. 




ca 



ffl 



Fig. 14. 



The sizes and pitches recommended are as follows 



TABLE XV. 

MACHINE SCREWS. 



Standard Diameter. 


.070 


.085 


.100 


.110 


.125 


.140 


.165 


.190 


.215 


.240 


.250 


.270 


.320 


.375 


Threads per inch. 


72 


64 


56 


48 


44 


40 


86 


32 


28 


24 


24 


22 


20 


16 



Eeference is made to the report itself for further 
details of heads, taps, etc. 

20. Eye Bolts and Hooks. In designing eye bolts 
it is customary to make the combined sectional area of 
the sides of the eye one and one half-times that of the 
bolt to allow for obliquity and an uneven distribution 
of stress. 

Large hooks should be designed to resist combined 



* Trans. A. S. M. E., Vol. xxvii. 



58 MACHINE DESIGN. 

bending and tension ; the bending moment is equal to 
the load multiplied by the longest perpendicular from 
the center line of hook to line of load. 

The tension due to this bending must be added to the 
direct tension and the body of the hook designed ac- 
cordingly. 

Problems. 

1. Discuss the effect of the initial tension caused by the 
screwing up of the nut on the bolt, in the case of steam fittings, 
etc. ; i. e. should this tension be added to the tension due to 
the steam pressure, in determining the proper size of bolt ? 

2. Determine the number of f inch steel bolts necessary to 
hold on the head of a steam cylinder 15 inches diameter, with 
the internal pressure 90 pounds per square inch, and factor of 
safety =12. 

3. Show what is the proper angle between the handle and 
the jaws of a fork wrench. * 

(1) If used for square nuts ; 

(2) If used for hexagon nuts ; illustrate by figure. 

4. Determine the length of nut theoretically necessary to 
prevent stripping of the thread, in terms of the outer diameter 
of the bolt. 

(1) With U. S. standard thread. 

(2) With square thread of same depth. 

5. Design a hook with a swivel and eye at the top to hold a 
load of one ton with a factor of safety 5, the center line of 
hook being three inches from line of load, and the material 
wrought iron. 

21. Riveted Joints. Kiveted joints may be divided 
into two general classes : lap joints where the two 
plates lap over each other, and butt joints where the 
edges of the plates butt together and are joined by 
over-lapping straps or welts. If the strap is on one 



RIVETED JOINTS. 



59 



side only, the joint is known as a butt joint with one 
strap ; if straps are used inside and out the joint is 
called a butt joint with two straps. Butt joints are 
generally used when the material is more than one 
half inch thick. 

Any joint may have one, two or more rows of rivets 
and hence be known as a single riveted joint, a double 
riveted joint, etc. 

Any riveted joint is 
weaker than the origi- 
nal plate, simply 
because holes cannot 
be punched or drilled 
in the plate for the 
introduction of rivets 
without removing 
some of the metal. 
Fig. 15 shows a 




A 







9 o 


MB 


D C _J 



Fig. 15. 



a line of 
at AB, 

of 



c 
c 



--<- 



) 



double riveted lap joint and Fig. 16 a single riveted 
butt joint with two straps. 

Eiveted joints may fail in any one of four ways : 

1. By tearing of the 
plate along 
rivet holes, as 
Fig. 15. 

2. By shearing 
the rivets. 

3. By crushing and 
wrinkling of the plate 
in front of each rivet 
as at C, Fig. 15, thus 
causing leakage. 

4. By splitting of the plate opposite each rivet as 
at D, Fig. 15. The last manner of failure may be pre- 



) 



o 


o 


o 


o 


o 


o < 





Fis:. 16. 



60 



MACHINE DESIGN. 



vented by having a sufficient distance from the rivet 
to the edge of the plate. Practice has shown that this 
distance should be at least equal to the diameter of a 
rivet. 

Experience has shown that lap joints in plates of 
even moderate thickness are dangerous on account of 
the liability of hidden cracks. Several disastrous 
boiler explosions have resulted from the presence of 
cracks inside the joint which could not be detected by 
inspection. The fact that one or both plates are out 
of the line of pull brings a bending moment on both 
plates and rivets. 

Some boiler inspectors have gone so far as to condemn 
lap-joints altogether. 

Let t= thickness of plate. 

d= diameter of rivet-hole. 

p= pitch of rivets. 

n= number of rows of rivets. 

T= tensile strength of plate. 

C = crushing strength of plate or rivet. 

S = Shearing strength of rivet. 
Average values of the constants are as follows : 



Material. 


T 


C 


s 


Wrought Iron 

Soft Steel 


50 000 
56 000 


80 000 
90 000 


40 000 
45 000 







The values of the 
average values and are 



constants given above are only 
liable to be modified by the 
exact grade of material used and the manner in which 
it is used. 



LAP JOINTS. 01 

The tensile strength of soft steel is reduced by 
punching from three to twelve per cent according to 
the kind of punch used and the width of pitch. The 
shearing strength of the rivets is diminished by their 
tendency to tip over or bend if they do not fill the 
holes, while the bearing or compression is doubtless 
relieved to some extent by the friction of the joint. 
The values given allow roughly for these modifica- 
tions. 

22. Lap Joints. This division also includes butt 
joints which have but one strap. 

Let us consider the shell as divided into strips at 
right angles to the seam and each of a width =p. 
Then the forces acting on each strip are the same and 
we need to consider but one strip. 

The resistance to tearing across of the strip between 

rivet holes is (p—d)tT. (a) 

and this is independent of the number of rows of 
rivets. 

The resistance to compression in front of rivets is 

ndtC (b) 

and the resistance to shearing of the rivets is 

\nd 2 S. (c) 

If we call the tensile strength T= unity then the 
relative values of C and S are 1.6 and 0.8 respectively. 

Substituting these relative values of T, C and S 
in our equations, by equating (b) and (c) and reducing 
we have d=2.55t (34) 

Equating (a) and (c) and reducing we have 

p=d+.Q2S^~. .... .(35) 



62 MACHINE DESIGN. 

Or by equating (a) and (b) 

p = d+l.Gnd (36) 

These proportions will give a joint of equal strength 
throughout, for the values of constants assumed. 

23. Butt Joints with two Straps. In this case the 
resistance to shearing is increased by the fact that the 
rivets must be sheared at both ends before the joint 
can give away. Experiment has shown this increase 
of shearing strength to be about 85 per cent and we 
can therefore take the relative value of S as 1.5 for 
butt joints. 

This gives the following values for d and p 

d=l.m (37) 

p=d+l.l&j- (38) 

p = d + l.Qnd (39) 

In the preceding formulas the diameter of hole and 
rivet have been assumed to be the same. 

The diameter of the cold rivet before insertion will 
be T V inches less than the diameter given by the 
formulas. 

Experiments made in England by Prof. Kennedy 
give the following as the proportions of maximum 
strength : 

Lap joints d=2.33t 

p=d+l.Z75nd 
Butt joints d=l.St 

p=±d+1.55nd 

24. Efficiency, of Joints. The efficiency of joints 
designed like the preceding is simply the ratio of the 
section of plate left between the rivets to the section 



BETT JOINIS WITH UNEQUAL STRAPS. 



63 



of solid plate, or the ratio of the clear distance between 
two adjacent rivet holes to the pitch. From formula 
(35) we thus have. 

Efficiency= I ^ (40) 

This gives the efficiency of single, double and triple 
riveted seams as 

.615, .762 and .828 respectively. 

Notice that the advantage of a double or triple 
riveted seam over the single is in the fact that the pitch 
bears a greater ratio to the diameter of a rivet, and 
therefore the proportion of metal removed is less. 

25. Butt Joints with unequal Straps. One joint in 
common use requires special treatment. 

It is a double-riveted butt joint in which the inner 
strap is made wider 
than the outer and 
an extra row of rivets 
added, whose pitch is 
double that of the 
original seam ; this is 
sometimes called 
diamond riveting. 
See Fig. 17. 

This outer row of 
rivets is then exposed 
to single shear and 
the original rows to 
double shear. 

Consider a strip of 
plate of a width = 2p. 





Fig. 17. 



Then the resistance to tearing along the outer row of 
rivets is (2p—cl)tT 



64 MACHINE DESIGN. 

As there are five rivets to compress in this strip the 
bearing resistance is 

6dtC 

As there is one rivet in single shear and four in 
double shear the resistance to shearing is 

1 + (4 X 1.85) 1 ^d 2 S = Q.6d 2 S 

Solving these equations as in previous cases, we 
have for this particular joint 

d=1.52t . (41) 

2p=9d 

p = 4=.5d (42) 

. . .(43) 



Efficiency = -^— = ( j 



26. Practical Rules. The formulas given above 
show the proportions of the usual forms of joints for 
uniform strength. 

In practice certain modifications are made for 
economic reasons. To avoid great variation iu the 
sizes of rivets the latter are graded by sixteenths of an 
inch, making those for the thicker plates considerably 
smaller than the formula would allow, and the pitch 
is then calculated to give equal tearing and shearing 
strength. 

Table XVI shows what may be considered average 
practice in this country for lap-joints with steel plates 
and rivets. 



PRACTICAL RULES. 



65 



TABLE XVI. 

RIVETED LAP JOINTS. 



Thick- 


Diam. 

of 
Rivet. 


Diam. 

of 
Hole. 


Pitch. 


Efficiency 


of Plate. 


ness of 
Plate. 


Single. 


Double. 


Single. 


Double. 


i 


1 


9 


If 


If 


.59 


.68 


A 


f 


tt 


If 


2i 


.58 


.68 


1 


t 


It 


11 


2i 


.57 


.67 


i 7 . 


13 


4 


2 


24 


.56 


.68 


i 


1 


if 


2 


21 


.53 


.67 



The efficiencies are calculated from the strength of 
plate between rivet holes and the efficiencies of the 
rivets may be even lower. Comparing these values 
with the ones given in Art. 24 we find them low. This 
is due to the fact that the pitches assumed are too 
small. The only excuse for this is the possibility of 
getting a tighter joint. 

TABLE XVII. 

RIVETED BUTT JOINTS. 



Thickness 


Diam. 

of 
Rivet. 


Diam. 

of 
Hole. 


Pitch. 


of 
Plate. 


Single. 


Double. 


Triple. 


i 


f 


« 


21 


4 


5*. 


1 


13 


1 


21 


3| 


5i 


1 


1 


H 


2| 


3| 


5i 


1 


H 


l 


2| 


8f 


5 


1 


1 


1A 


2f 


8* 


5 



6Q MACHINE DESIGN. 

Table XVII has been calculated for butt joints with 
two straps. As in the preceding table the values of 
the pitch are too small for the best efficiency. The 
tables are only intended to illustrate common practice 
and not to serve as standards. There is such a diversity 
of practice among manufacturers that it is advisable 
for the designer to proportion each joint according to 
his own judgment, using the rules of Arts. 22-25 and 
having regard to the practical considerations which 
have been mentioned. 

A committee of the Master Steam Boiler Makers' 
Association has made a number of tests on riveted 
joints and reported its conclusions. The specimens 
were prepared according to generally accepted practice, 
but on subjecting them to tension many of them failed 
by tearing through from hole to edge of plate. The 
committee recommends making this distance greater, 
so that from the center of hole to edge of plate shall be 
perhaps 2d instead of 1.5d. 

The committee further found the shearing strength 
of rivets to be in pounds per square inch of section. 





Single Shear. 


Double Shear. 


Iron rivets 

Steel rivets 


40000 
49000 


78000 
84000 







Compare these values with those given in Art. 21. 
Also note that the factor for double shear is 1.95 for 
iron rivets and only 1.71 for steel rivets as against the 
1.85 given in Art. 23. The committee found that 
machine-driven rivets were stronger in double shear 
than hand-driven ones. 



PRACTICAL RULES. 
Problems. 



67 



1. Calculate diameter and pitch of rivets for -*- in. and 
i in. plate and compare results with those in Table XVI. 
Criticise latter. 

2. Show the effect in Prob. 1 of using iron rivets in steel 
plates. 

3. Criticise proportions of joints for i in. and 1 in. plate in 
Table XVII. by testing the efficiency of rivets and plates. 

4. A cylinder boiler 5X16 ft. is to have long seams double- 
riveted laps and ring seams single riveted laps. If the inter- 
nal pressure is 90 lbs. gauge pressure and the material soft 
steel, determine thickness of plate and proportion of joints. 
The net factor of safety at joints to be five. 

5. A marine boiler is 11 ft. 6 ins. in diameter and 14 ft long. 
The long seams are to be diamond riveted butt joints and the 
ring seams ordinary double riveted butt joints. The internal 
pressure is to be 175 lbs. gauge and the material is to be steel 
of 60,000 lbs. tensile strength. Determine thickness of shell 
and proportions of joints. Net factor of safety to be 5, as in 
Prob. 4. 

6. Design a diamond riveted joint such as shown in Fig 18 
for a steel plate -| in. thick. Outer cover plate is -| in. and 
inner cover plate is T 7 ^ inches thick ; the pitch of outer rows of 
rivets to be twice that of inner rows. Determine efficiency of 
joints. 




7. The single lap joint with cover plate, as shown in Fig. 19, 
is to have pitch of outer rivets double that of inner row. De- 



68 MACHINE DESIGN. 

termine diameter and pitch of rivets for | inch plate and the 
efficiency of joint. 



' ""■- \w//A w 



\_7 




Fig. 19. 



27. Riveted Joints for Narrow Plates. The joints 
which have been so far described are continuous and 
but one strip of a width equal to the pitch or the least 
common multiple of several pitches, has been investi- 
gated. 

When narrow plates such as are used in structural 
work are to be joined by rivets, the joint is designed 
as a whole. Diamond riveting similar to that shown 
in Fig. 17 is generally used and the joint may be a 
lap, or a butt with double straps. The diameter of 
rivets may be taken about 1.5 times the thickness of 
plate [see equation (41) ], and enough rivets used so 
that the total shearing strength may equal the tensile 
strength of the plate at the point of the diamond, 
where there is one rivet hole. It may be necessary to 
put in more rivets of a less diameter in order to make 
the figure symmetrical. 

The efficiency of the joint may be tested at the dif- 
ferent rows of rivets, allowing for tension of plate 
and shear of rivets in each case. 

Problems. 

1. Design a diamond-riveted lap-joint for a plate ten inches 
wide and one-half inch thick, and calculate least efficiency for 
shear and tension. 



JOINT PINS. 



69 



1 — ~ — " 


o 


o o 


o o 

O O 


o o 


o 


_x~>-— J 



2. A diamond-riveted butt-joint with two straps has rivets 
arranged as in Fig. 20, the plate being twelve 
inches wide and three-quarter inches thick, 
and the rivets being one inch in diameter. If 
the plate and rivets are of steel, find the 
probable ultimate strength of the following 
parts : 

(a) The whole plate. 

(b) All the rivets on one side of the joint. 

(c) The joint at the point of the dia- 
mond. 

(d) The joint at the row of rivets next 
the point. 

Fig. 20. 

28. Joint Pins. A joint pin is a bolt exposed to 
double shear. If the pin is loose in its bearings it 
should be designed with allowance for bending, by 
adding from 30 to 50 per cent to the area of cross- 
section needed to resist shearing alone. Bending of the 
pin also tends to spread apart the bearings and this 
should be prevented by having a head and nut or cotter 
on the pin. 

If the pin is used to connect a knuckle joint as in 
boiler stays, the eyes forming the joint should have a 
a net area 50 per cent in excess of the body of the stay, 
to allow for bending and uneven tension, (see Eye- 
bolts, Art. 20.) 

Fig. 21 shows a pin and angle joint for attaching 
the end of a boiler stay to the head of the boiler. 



29. Cotters. A cotter is a key which passes diamet- 
rically through a hub and its rod or shaft, to fasten 



70 



MACHINE DESIGN. 



them together, and is so called to distinguish it from 
shafting keys which lie parallel to axis of shaft. 



c 



n 



^w 





Fig. 21. 



Fig. 22. 



Its taper should not be more than 4 degrees or 
about 1 in 15, unless it is secured by a screw or check 
nut. 

The rod is sometimes enlarged where it goes in the 
hub, so that the effective area of cross-section where 
the cotter goes through may be the same as. in the 
body of the rod. (See Fig. 22.) 

Let : d = diameter of body of rod. 

d = diameter of enlarged portion. 

t= thickness of cotter, usually 3 

6= breadth of cotter. 
1= length of rod beyond cotter. 
Suppose that the applied force is a pull on the rod — 
causing tension on the rod and shearing stress on the 
cotter. 

The effective area of cross section of rod at cotter is 

4 4 



A 



=(" 



rf 2 

i)4 



COTTERS. ft 

and this should equal the area of cross-section of the 
body of rod. 

^=^-^--=1.21^ (44) 

7T 1 

Let P=pull on rod. 

S= shearing strength of material. 
The area to resist shearing, of cotter is 

2t>t 2 ^ 

• • &= d^ < a ) 

The area to resist shearing of rod is 
p 

andZ= 3335 (b) 

If the metal of rod and cotter are the same 

l=\ («) 

Great care should be taken in fitting cotters that 
they may not bear on corners of hole and thus tear the 
rod in two. 

A cotter or pin subjected to alternate stresses in op- 
posite directions should have a factor of safety double 
that otherwise allowed. 

Adjustable cotters, used for tightening joints of 
bearings are usually accompanied by a gib having a 



MACHINE DESIGN. 



£. 



XI 




taper equal and opposite to that of the cotter. (Fig. 

23). In designing these for 
strength the two can be re- 
garded as resisting shear to- 
gether. 

For shafting keys see chapter 
on shafting. 

The split pin is in the nature 
of a cotter but is not usually- 
expected to take any shearing 
stress. 



Fig. 23. 



Problems. 



1. Design an angle joint for a soft steel boiler stay, the pull 
on stay being 12000 lbs. and the factor of safety, six. Use 
two standard angles. 

2. Determine the diameter of a round cotter pin for equal 
strength of rod and pin. 

3. A rod of wrought iron has keyed to it a piston 18 inches 
in diameter, by a cotter of machinery steel. 

Eequired the two diameters of rod and dimensions of cotter 
to sustain a pressure of 150 pounds per square inch on the 
piston. Factor of safety = 8. 

Design a cotter and gib for connecting rod of engine men- 
tioned in Prob. 3, both to be of machinery steel and .75 inches 
thick. (See Fig. 23.) 



CHAPTER V. 

SPRINGS. 

30. Helical Springs. The most common form of 
spring used in machinery is the spiral or helical spring 
made of round brass or steel wire. Such springs may 
be used to resist extension or compression or they may 
be. used to resist a twisting moment. 

Tension and Compression* 

Let L= length of axis of spring. 
Z)=mean diameter of spring. 
1= developed length of wire. 
d= diameter of wire. 

R= ratio - T . 
d 

n= number of coils. 

P= tensile or compressive force. 

x= corresponding extension or compression. 

iS^safe torsional or shearing strength of wire. 

=45000 to 60000 for spring brass wire. 

= 75000 to 115000 for cast steel tempered. . 
G= modulus of torsional elasticity. 

= 6000000 for spring brass wire. 

= 12000000 to 15000000 for cast steel, tempered. 

Then l=V^D 2 n 2J rU 

If the spring were extended until the wire became 

73 



^4: MACHINE DESIGN. 

straight it would then be twisted n times, or through 
an angle =2im and the stretch would be I — L. 
The angle of torsion for a stretch =x is then 

A %mx 



l—L 



(a) 



Suppose that a force P' acting at a radius-^ will 

twist this same piece of wire through an angle causing 
a stress S at the surface of the wire. Then will the 
distortion of the surface of the wire per inch of length 

be s=~ and the stress S=^~ = 5A ^ D ...(b) 

- G= J = zero (c) 

In thus twisting the wire the force required will 
vary uniformly from o at the beginning to P' at the 
end provided the elastic limit is not passed, and the 
average force will be 

P' P f DB 

=— The work done is therefore — : — ■ 

If the wire is twisted through the same angle by 
the gradual application of the direct pressure P, com- 
pressing or extending the spring the amount x, the 
work done will be , 

Px -d >P'DQ_Px 

Substituting this value of P' in (c) and solving for x : 



x= 



10.2 PI 



SQUARE WIRE. 75 

Substituting the value of from (a) and again solving 
for T : 

^_10^2Pl ( l—L I 2 , v 

lT — /a^ 1 aw, f * ♦ \ e ) 



Gd* { 2tt)i 

If we neglect the original obliquity of the wire then 
l=irDn and L=o and equation (e) reduces to 

x ~ Gd* { } 

Making the same approximation in equation (d) we 
have P'=P 

i.e. — a force P will twist the wire through approxi- 
mately the same angle when applied to extend or com- 
press the spring, as if applied directly to twist a piece 
of straight wire of the same material with a lever 

D 

arm^-^r- 

This may be easily shown by a model. 
The safe working load may be found by solving for 
P' in (b) and remembering that P = P' 

r, Sd 3 Sd 2 



2.55D~2,55i? 



(47) 



when S is the safe shearing strength ♦ 

Substituting this value of P in (32) we have for the 
safe deflection : 

„_]DS IRS x 4S n 

31. Square Wire. The value of the stress for a 
square section is : 

where d is the side of square. 



76 MACHINE DESIGN. 

The distortion at the corners caused by twisting 
through an angle 6 is : 

Bd 





S V2 


Equation 


(c) then becomes : 




r GP'Dl 

U= 2d 4 


The three 
reduce to : 


principal equations (46), (47) and (48) then 




x ~ Gd* ^ 




p -£i D ( 5 °> 




x- WS _ (51) 



Gdi/2 

The square section is not so economical of material 
as the round, 

32. Experiments. Tests made on about 1700 tem- 
pered steel springs at the French Spring Works in 
Pittsburg were reported in 1901 by Mr. K. A. French.* 
These were all compression springs of round steel and 
were given a permanent set before testing by being 
closed coil to coil several times. Table XVIII gives 
results of these experiments. 

* Trans. A. S. M. E., Vol. XXIII. 



EXPERIMENTS. 



77 



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Y8 MACHINE DESIGN. 

The apparent variation of G in the experiments is 
probably due to differences in the quality of steel and 
to the fact that the formula for G in the case of helical 
springs is an approximate one. 

The same may be said of the values of S, but if these 
values are used in designing similar springs one error 
will off-set the other. 

In some few cases, as in No. 18, it was necessary to 
use an abnormally high value of S to meet the condi- 
tions. This necessitated a special grade of steel, and 
great care in manufacture. Such a spring is not safe 
when subjected to sudden and heavy loads, or to rapid 
vibrations, as it would soon break under such treat- 
ment ; if merely subjected to normal stress, it would 
last for years. 

Springs of a small diameter may safely be subjected 
to a higher stress than those of a larger diameter, the 
size of bar being the same. The safe variation of S 
with R cannot yet be stated. 

There is an important limit which should be here 
mentioned. Springs having too small a diameter as 
compared with size of bar are subjected to so much 
internal stress in coiling as to weaken the steel. A 
spring, to give good service, should never have B less 
than 3. 

The size of bar has much to do with the safe value 
of S ; the probable explanation is this : A large bar 
has to be heated to a higher temperature in working 
it, and in high carbon steel this may cause deterio- 
ration ; when tempered, the bath does not affect it so 
uniformly, as may be seen by examining the fracture 
of a large bar. 

The above facts must always be taken into consid- 
eration in designing a spring, whatever the grade of 



SPRING IN TORSION. 79 

steel used. A safe value of & can be determined only 
by one having an accurate knowledge of the physical 
characteristics of the steel, the proportions of the 
spring, and the conditions of use. 

7. For a good grade of steel the following values of 
S have been found safe under ordinary conditions of 
service, the value of G being taken as 14,500,000. 

VALUES OF S. 



d=% inch or less 

d=^ inch to f inch. 
d={l inch to 1J inch 



R=S 



112,000 
110,000 

105,000 



R-. 



85,000 
80,000 
75.000 



For bars over 1J inches in diameter a stress of more 
than 100,000 should not be used. Where a spring is 
subjected to sudden shocks a smaller value of $ is 
necessary. 

As has been noted, the springs referred to in this 
paper were all compression springs. Experience has 
shown that in close coil or extension springs the value 
of G is the same, but that the safe value of: S is only 
about two-thirds that for a compression spring of the 
same dimensions. 

33. Spring in Torsion. If a helical spring is used to 
resist torsion instead of tension or compression, the 
wire itself is subjected to a bending moment. We will 
use the same notation as in the last article, only that 
P will be taken as a force acting tangentially to the 

circumference of the spring at a distance -^- from the 



80 MACHINE DESIGN. 

axis, and S will now be the safe transverse strength of 
the wire, having the following values : 
S = 60, 000 for spring brass wire. 

= 90,000 to 125000 for cast steel tempered. . 
i£ = 9000000 for spring brass wire. 
= 30000000 for cast steel tempered. 
Let 6= angle through which the spring is turned 
by P. 

The bending moment on the wire will be the same 

PD 

throughout and =-^— . This is best illustrated by a 

model. 

To entirely straighten the wire by unwinding the 
spring would require the same force as to bend straight 
wire to the curvature of the helix. 

To simplify the equations we will disregard the 
obliquity of the helix, then will 1=ttDji and the radius 

of curvature = w- 

Let M= bending moment caused by entirely 
straightening the wire ; then by mechanics 

EI 2EI 



M-- 



B D 



and the corresponding angle through which spring is 
turned is 2im. 

But it is assumed that a force P with a radius — ~- 

turns the spring through an angle 0. 

2 ~ D X 2vm 

_EI0_E16 
~wDn~ I 



FLAT SPRINGS. 81 

Solving for 6 : 

« PDl 



and if wire is round 



(a) 



•-^ w 



The bending moment for round wire will be 

T io (53) 

and this will also be the safe twisting moment that can 
be applied to the spring when S= working strength of 
wire. The safe angle of deflection is found by sub- 

PD 

stituting this value of -^r- in (52): 

Eeducing: *=§§■ •' • (54) 

34. Flat Springs. Ordinary flat springs of uniform 
rectangular cross-section can be treated as beams and 
their strength and deflection calculated by the usual 
formulas. 

In such a spring the bending and the stress are 
greatest at some one point and the curvature is not 
uniform. 

To correct this fault the spring is made of a constant 
depth but varying width. 

If the spring is fixed at one end and loaded at the 
other the plan should be a triangle with the apex at 
loaded end. If it is supported at the two ends and 
loaded at the center, the plan should be two triangles 
with their bases together under the load forming a 




82 MACHINE DESIGN. 

rhombus. The deflection of such a spring is one and 
a half times that of a rectangular spring. 

As such a spring 
might be of an in- 
convenient width, 
a compound or leaf- 
spring is made by 
cutting the trian- 
gular spring into 
strips parallel to the 
axis, and piling one 
Fi s- 34 - above another as in 

Fig. 24. 
This arrangement does not change the principle, 
save that the friction between the leaves may increase 
the resistance somewhat. 

Let 1= length of span. 
b= breadth of leaves. 
t= thickness of leaves. 
n= number of leaves. 
FF=load at center. 
A = deflection at center. 
& and E may be taken as 80000 and 30000000 re- 
spectively. 
Strength : 

M Wl Snbt 2 
4 ~ 6 

W=l-^ (55) 



Elasticity : 



. Wf , T nhf 

A = 32iTl where 7= T2- 

A =8EnbP' •■•■•• .W 




ELLIPTIC AND SEMI-ELLIPTIC SPRINGS. 83 

35. Elliptic and Semi- Elliptic Springs. Springs as 
they are usually designed for service differ in some re- 
spects from those 
just described, as 
may be seen by re- 
ference to Fig. 25. 
A band is used at 
the center to con- 
fine the leaves in Fig. 25. 
place. As this band 

constrains the spring at the center it is best to con- 
sider the latter as made up of two cantilevers each 

having a length of -g— where w is the width of band. 

The spring usually contains several full-length leaves 
with blunt ends, the remaining leaves being graduated 
as to length and pointed as in Fig. 25. The blunt 
full-length leaves constitute cantilevers of uniform 
cross-section, while the graduated leaves form canti- 
levers of uniform strength. Under similar conditions 
as to load and fiber stress the latter will have a deflec- 
tion fifty per cent greater than the former. Suppos- 
ing that there is no initial stress between the leaves 
caused by the band, both sets must have the same de- 
flection. This means that more than its proportion of 
the load will be carried by the full-length set and con- 
sequently it will have a greater fiber stress. This 
difficulty can be obviated by having an initial gap be- 
tween the graduated set and the full-length set and 
closing this with the band. 

If this gap is made half the working deflection of 
the spring, the total deflection of the graduated set 
under the working load will be fifty per cent greater 
than that of the full-length set and the fiber stress will 
be uniform. 



84 MACHINE DESIGN. 

The load will then be divided between the two sets 
in proportion to the number of leaves in each. 

One of the full-length leaves must be counted as a 
part of the graduated set. When the gap is closed by 
a band there will be an initial pull on the band due to 
the deflection of the spring. 

This can be determined for any given spring by re- 
garding the two sets of leaves as simple beams the 
sum of whose deflections under the pull P is equal to 
the depth of the gap. 

Full elliptic springs can be designed in a similar 
manner but the total deflection will be double that of 
the semi-elliptic spring. 

PBOBLEMS. 

1. A spring balance is to weigh 25 pounds with an extension 
of 2 inches, the diameter of spring being § inches and the 
material, tempered steel. 

Determine the diameter and length of wire, and number of 
coils. 

2. Determine the safe twisting moment and angle of torsion 
for the spring in example 1, if used for a torsional spring. 

3. Test values of G and S from data given in Table XVIII. 

4. By using above table design a spring 8 ins. long to carry 
a load of 2 tons without closing the coils more than half way. 

5. Design a compound flat spring for a locomotive to sustain 
a load of 16000 lbs. at the center, the span being 40 inches, the 
number of leaves 12 and the material steel. 

6. Determine the maximum deflection of the above spring, 
under the working load. 

7. A semi-elliptic spring has N leaves in all and n graduated 

leaves, and the load on each end is P=- ir . Develop formulas 

for the fiber stress in each set of leaves if there is no initial 
stress. 



ELLIPTIC AND SEMI-ELLIPTIC SPRINGS. 85 

8. In Prob. 7 develop a formula for the necessary gap to 
equalize the fiber stresses. 

9. In Prob. 8 determine the pull on the band due to the 
initial stress. 

10. A semi-elliptic spring has 4 leaves 36 inches long, and 12 
graduated leaves. The leaves are all 4 inches wide and | 
inches thick, and the band at the center is 4 inches wide. If 
there is no initial stress find the share of the load and the fiber 
stress on each set of leaves when there is a load of 6 tons on 
the center. Also determine deflection. 

11. In Prob. 10, determine the amount of gap needed to 
equalize the stresses in the two sets of leaves, and the pull on 
the band at the center. Determine the deflection under the 
load. 

12. Measure various indicator springs and determine value 
of G from rating of springs. 

13. Measure various brass extension springs, calculate safe 
static load and safe stretch. 

14. Make an experiment on torsion spring to determine dis- 
tortion under a given load and calculate value of E. 



CHAPTER VI. 

SLIDING BEARINGS. 

36. Slides in General. The surfaces of all slides 
should have sufficient area to limit the intensity of 
pressure and prevent forcing out of the lubricant. No 
general rule can be given for the limit of pressure. 
Tool marks parallel to the sliding motion should not 
be allowed, as they tend to start grooving. The sliding 
piece should be as long as practicable to avoid local 
wear on stationary piece and for the same reason should 
have sufficient stiffness to prevent springing. A slide 
which is in continuous motion should lap over the 
guides at the ends of stroke, to prevent the wearing of 
shoulders on the latter and the finished surfaces of all 
slides should have exactly the same width as the sur- 
faces on which they move for a similar reason. 

Where there are two parallel guides to motion as in 
a lathe or planer it is better to have but one of these 
depended upon as an accurate guide and to use the 
other merely as a support. It must be remembered 
that any sliding bearing is but a copy of the ways of 
the machine on which it was planed or ground and in 
turn may reproduce these same errors in other ma- 
chines. The interposition of handscraping is the only 
cure for these hereditary complaints. 

In designing a slide one must consider whether it is 
accuracy of motion that is sought, as in the ways of 
a planer or lathe, or accuracy of position as in the head 

86 



ANGULAR SLIDES. 



87 



of a milling machine. Slides may be divided accord- 
ing to their shapes into angular, flat and circular slides. 

37. Angular Slides. An angular slide is one in 
which the guiding surface is not normal to the direc- 
tion of pressure. There is a tendency to displacement 
sideways, which necessitates a second guiding surface 
inclined to the first. This oblique pressure constitutes 
the principal disadvantage of angular slides. Their 
principal advantage is the fact that they are either 
self-adjusting for wear, as in the ways of lathes and 
planers, or require at most but one adjustment. 

Fig. 26 shows one of the 
Vs of an ordinary planing 
machine. The platen is held 
in place by gravity. The 
angle between the two sur- 
faces is usually 90 deg. but 
may be more in heavy ma- 
chines. The grooves g, g are 
intended to hold the oil in 
place ; oiling is sometimes 
effected by small rolls recessed into the lower piece 
and held against the platen by springs. 

The principal advantage 
of this form of way is its 
ability to hold oil and the 
great disadvantage its fa- 
culty for catching chips and 
dirt. 

Fig 27 shows an inverted 
V such as is common on the 
ways of engine lathes. The 
angle is about the same as in the preceding form but 




Fig. 26. 




Fig. 27 



88 



MACHINE DESIGN. 



the top of the V should be rounded as a precaution 
against nicks and bruises. 

The inverted V is preferred for lathes since it will 
not catch dirt and chips. It needs frequent lubrica- 
tion as the oil runs off rapidly. Some lathe carriages 
are provided with extensions filled with oily felt or 
waste to protect the ways from dirt and keep them 
wiped and oiled. Side pressure tends to throw the 
carriage from the ways ; this action may be prevented 
by a heavy weight hung on the carriage or by gibbing 
the carriage at the back. (See Fig. 33). The objection 
to this latter form of construction is the fact that it is 
practically impossible to make and keep the two F's 
and the gibbed slide all parallel. 

Fig. 28 shows a compound 
V sometimes used on heavy 
machines. The obtuse 
angle (about 150 deg.) takes 
the heavy vertical pressure, 
while the sides, inclined 
only 8 or 10 deg., take any 
side pressure which may 
develop. 




Fig. 28. 



38. Gibbed Slides. All slides which are not self- 
adjusting for wear must be provided with gibs and 
adjusting screws. Fig. 29 
shows the most common 
form as used in tool slides 
for lathes and planing 
machines. 

The angle employed is 
usually 60 deg. ; notice that 
the corners c c are clipped for strength and to avoid a 




Fig. 29. 



FLAT SLIDES. 



89 



corner bearing ; notice also the shape of gib. It is bet- 
ter to have the points of screws coned to fit gib and 
not to have flat points fitting recesses in gib. The 
latter form tends to spread joint apart by forcing gib 
down. If the gib is too thin it will spring under the 
screws and cause uneven wear. 

The cast iron gib, Fig. 30, 
is free from this latter defect 
but makes the slide rather 
clumsy. The screws how- 
ever are more accessible in 
this form. Gribs are some- 
times made slightly tapering 
and adjusted by a screw and 



-po. 



r.! J 



Fig. 30. 



nut giving endwise motion. 



39. Flat Slides. This type of slide requires adjust- 
ment in two directions and is usually provided with 
gibs and adjusting screws. Flat ways on machine 
tools are the rule in English practice and are gradually 
coming into use in this country. Although more ex- 
pensive at first and not so simple they are more durable 
and usually more accurate than the angular ways. 

Fig. 31 illustrates a flat way for a planing machine. 
The other way would 
be similar to this but 
without adjustment. 
The normal pressure 
and the friction are 
less than with angular 
ways and no amount 
of side pressure will 
lift the platen from its 
position. Fig. 31. 




90 



MACHINE DESIGN. 



Fig. 32 shows a portion of the ram of a shaping 
machine and illustrates the use of an L gib for adjust- 
, • — 1 ment in two directions. 



C jTJtrmT 




Fig. 33 shows a gibbed 
slide for holding down 
the back of a lathe car- 
riage with two adjust- 
ments. 

The gib g is tapered 
and adjusted by a screw 
and nuts. The saddle 
of a planing machine or 
the table of a shaper usually has a rectangular gibbed 



Fig. 32. 





Fig. 33. 



Fig. 34. 



slide above and a taper slide below, this form of the 
upper slide being necessary to hold the weight of the 
overhanging metal. (See Fig. 34.) Some lathes and 
planers are built with one V or angular way for 
guiding the carriage of platen and one flat way acting 
merely as a support. 



STUFFING BOXES. 



91 



40. Circular Guides. Examples of this form may 
be found in the column of the drill press and the over- 
hanging arm of the 

The A 



B 



milling machine 
cross heads of steam 
engines are sometimes 
fitted with circular 
guides ; they are more 
frequently flat or an- 
gular. One advantage 
of the circular form is 
the fact that the cross 
head can adjust itself 
to bring the wrist pin parallel to the crank pin. The 
guides can be bored at the same setting as the cylinder 
in small engines and thus secure good alignment. 

Fig. 35 illustrates various shapes of cross head slides 
in common use. 





Fig. 35. 



41. Stuffing Boxes. In steam engines and pumps 
the glands for holding the steam and water packing 
are the sliding bearings which cause the greatest fric- 




Fig. 36. 

tion and the most trouble. Fig. 36 shows the general 



92 MACHINE DESIGN. 

arrangement. B is the stuffing box attached to the 
cylinder head ; R is the piston rod ; G the gland ad- 
justed by nuts on the studs shown ; P the packing 
contained in a recess in the box and consisting of rings, 
either of some elastic fibrous material like hemp and 
woven rubber cloth or of some soft metal like babbit. 
The pressure between the packing and the rod, neces- 
sary to prevent leakage of steam or water, is the cause 
of considerable friction and lost work. Experiments 
made from time to time in the laboratories of the Case 
School have shown the extent and manner of variation 
of this friction. The results for steam packings may 
be summarized as follows : 

1. That the softer rubber and graphite packings, 
which are self-adjusting and self -lubricating, as in 
Nos. 2, 3, 7, 8, and 11, consume less power than the 
harder varieties. No. 17, the old braided flax style, 
gives very good results. 

2. That oiling the rod will reduce the friction with 
any packing. 

3. That there is almost no limit to the loss caused 
by the injudicious use of the monkey-wrench. 

4. That the power loss varies almost directly with 
the steam pressure in the harder varieties, while it is 
approximately constant with the softer kinds. 

The diameter of rod used — two inches — would be ap- 
propriate for engines from 50 to 100 horse-power. The 
piston speed was about 140 feet per minute in the 
experiments, and the horse-power varied from .036 to 
.400 at 50 pounds steam pressure, with a safe average 
for the softer class of packings of .07 horse- 
power. 

At a piston speed of 600 feet per minute, the same 
friction would give a loss of from .154 to 1.71 with a 



STUFFING BOXES. 



93 



working average of .30 horse-power, at a mean steam 
pressure of 50 pounds. 

In Table XIX Nos. 6, 14, 15 and 16 are square, hard 
rubber packings without lubricants. 

Similar experiments on hydraulic packings under 
a water pressure varying from ten to eighty pounds 
per square inch gave results as shown in Table 
XXI. 

The figures given are for a two inch rod running 
at an average piston speed of 140 feet per minute. 

TABLE XIX. 





3 
H 

o 


Total Time 
of Run in 
Minutes. 


Av. Horse 
Power Con- 
sumed by 
each Box. 


Horse Pow. 
Cons, at 50 
Lbs. Press. 


i 


5 


22 


.091 


.085 


2 


8 


40 


.049 


.048 


3 


5 


25 


.037 


.036 


4 


5 


25 


.159 


.176 


5 


5 


25 


.095 


.081 


6 


5 


25 


.368 


.400 


7 


5 


25 


.067 


.067 


8 


5 


25 


.082 


.082 


9 


3 


15 


.200 


.182 


10 


3 




.275 




11 


5 


25 


.157 


.172 


12 


5 


25 


.266 


.330 


13 


5 


25 


.162 


.230 


14 


5 


25 


.176 


.276 


15 


5 


25 


.233 


.255 


16 


5 


25 


.292 


.210 


17 


5 


25 


.128 


.084 



Remarks on Leakage, etc. 



Moderate leakage. 

Easily adjusted ; slight leakage. 

Considerable leakage. 

Leaked badly. 

Oiling necessary ; leaked badly. 

Moderate leakage. 

Easily adjusted and no leakage. 

Very satisfactory ; slight leakage. 

Moderate leakage. 

Excessive leakage. 

Moderate leakage. 

Moderate leakage. 

No leakage ; oiling necessary. 

Moderate leakage ; oiling necessary. 

Difficult to adjust ; no leakage. 

Oiling necessary ; no leakage. 

No leakage. 



94 



MACHINE DESIGN. 
TABLE XX. 



to 

.5 

O 
©PL, 

B 



1 

3 

4 

5 

6 

7 

8 

9 

11 

12 

13 

15 

16 

17 



Horse Power consumed by each Box, when 

Pressure was applied to Gland Nuts 

by a Seven-Inch Wrench. 



5 

Pounds. 



120 



8 


10 


12 


14 


Pounds. Po 


unds. Po 


unds. 


Pounds. 


.... 


136 






!248 




303 


.... 


.220 








.348 


430 






.126 


228 


260 


.330 


.363 


500 


535 


.520 


.666 








.405 


454 






.161 


242 


359 


.454 


.317 


394 


582 




.526 








.327 


860 






.198 


277 


380 





16 

Pounds. 



390 



340 
533 



o <£:^ • 



W 



Dry. 



055 
154 

323 
067 
533 
666 
454 
454 



Oiled. 



021 
123 

194 
053 
236 
636 
176 
122 



TABLE XXI. 



No. of 


Av. H. P. 


Av. H. P. 


Max. 


Min. 


Av. H. P. 




at 


at 






for entire 


Packing. 


20 lbs. 


70 Lbs. 


H. P. 


H. P. 


Test. 


1 


.077 


.351 


.452 


.024 


.259 


2 


.422 


.500 


.512 


.167 


.410 


3 


.130 


.178 


.276 


.035 


.120 


4 


.184 


.195 


.230 


.142 


.188 


5 


.146 


.162 


.285 


.069 


.158 


6 


.240 


.200 


.255 


.071 


.186 


7 


.127 


.192 


.213 


.095 


.154 


8 


.153 


.174 


.238 


.112 


.165 


9 


.287 


.469 


.535 


.159 


.389 


10 


.151 


.160 


.226 


.035 


.103 


11 


.141 


.156 


.380 


.064 


.177 


12 


,053 


.095 


.143 


.035 


.090 



STUFFING BOXES. 95 

Packings Nos. 5, 6, 10 and 12 are braided flax with 
graphite lubrication and are best adapted for low 
pressures. Packings Nos. 3, 4 and 7 are similar to the 
above but have paraffine lubrication. Packings Nos. 
2 and 9 are square duck without lubricant and are only 
suitable for very high pressures, the friction loss being 
approximately constant. 

PROBLEMS. 

Make a careful study and sketch of the sliding bearings on 
each of the following machines and analyze as to (a) Purpose 
(6) Character, (c) Adjustment, (d) Lubrication. 

1. An engine lathe. 

2. A planing machine. 

3. A shaping machine. 

4. A milling machine. 

5. An upright drill. 

6. A Corliss engine. 

7. A Porter- Allen engine. 

8. A gas-engine. 

9. An air-compressor. 



CHAPTER VII. 



JOURNALS, PIVOTS AND BEARINGS. 

42. Journals. A journal is that part of a rotating 
shaft which rests in the bearings and is of necessity a 
surface of revolution, usually cylindrical or conical. 
The material of the journal is generally steel, some- 
times soft and sometimes hardened and ground. 

The material of the bearing should be softer than 
the journal and of such a quality as to hold oil readily. 
The cast metals such as cast iron, bronze and babbit 
metal are suitable on account of their porous, granular 
character. Wood, having the grain normal to the 
bearing surface, is used where water is the lubricant, 
as in water wheel steps and stern bearings of propellers. 

43. Adjustment. Bearings wear more or less rapidly 
with use and need to be adjusted to compensate for 

the wear. The adjust- 
ment must be of such a 
character and in such a 
direction as to take up the 
wear and at the same 
time maintain as far as 
possible the correct shape 
of the bearing. The ad- 
justment should then be 
in the line of the greatest 
pressure. 

Fig. 37 illustrates some 
of the more common ways of adjusting a bearing, the 




ADJUSTMENT. 



97 



arrows showing the direction of adjustment and presu- 
mably the direction of pressure ; (a) is the most usual 
where the principal wear is vertical, (d) is a form 
frequently used on the main journals of engines when 
the wear is in two directions, horizontal on account of 
the steam pressure and vertical on account of the 
weight of shaft and fly wheel. All of these are more 
or less imperfect since the bearing, after wear and ad- 
justment, is no longer cylindrical but is made up of 
two or more approximately cylindrical surfaces, 

A bearing slightly 
conical and adjusted 
endwise as it wears, is 
probably the closest 
approximation to cor- 
rect practice ♦ 

Fig. 38 shows the 
main bearing of the 
Porter - Allen engine, 
one of the best ex- 
amples of a four part ad justment* The cap, is adjusted 

in the normal way with 
bolts and nuts ; the bot- 
tom, can be raised and 
lowered by liners placed 
underneath ; the cheeks 
can be moved in or 
out by means of the 
wedges shown* Thus it 
is possible, not only to 
adjust the bearing for 
wear, but to align the 
shaft perfectly. 
Fig. 39. A three part bearing 




Fig. 38. 




98 



MACHINE DESIGN. 



for the main journal of an engine is shown in Fig. 
39. In this bearing there is one horizontal adjust- 
ment, instead of two as in Fig. 38. 

The main bearing of the 
spindle in a lathe, as shown in 
Fig. 40, offers a good example 
of symmetrical adjustment. 
The headstock A has a conical 
hole to receive the bearing B, 
which latter can be moved 
lengthwise by the nuts F G. 
The bearing may be split into 
Fig. 40. two, three or four segments or 

it may be cut as shown in (e) 
Fig. 37, and sprung into adjustment. A careful 
distinction must be made between this class of bearing 
and that before mentioned, where the journal itself is 
conical and adjusted endwise. A good example of 
the latter form is seen in the spindles of many milling 
machines. 





Fig. 41. 

Fig. 41 shows the spindle of an engine lathe complete 
with its two bearings. The end thrust is taken by a 
fiber washer backed by an adjusting collar and check 
nut. Both bearings belong to the class shown in 
Fig. 40. 

A conical journal with end adjustment is illustrated 



LUBRICATION. 



99 



in Fig, 42, which shows the spindle of a milling 
machine* The front journal is conical and is adjusted 




Fig. 42. 

for wear by drawing it back into its bearing with the 
nut. The rear journal on the other hand is cylindrical 
and its bearing is adjusted as are those just described. 
The end thrust is taken by two loose rings at the front 
end of the spindle. 

44. Lubrication. The bearings of machines which 
run intermittently, like most machine tools, are oiled 
by means of simple oil holes, but machinery which is 
in continuous motion as is the case with line shafting 
and engines requires some automatic system of lubri- 
cation. There is not space in this book for a detailed 
description of all the various types of oiling devices 
and only a general classification will be attempted. 

Lubrication is effected in the following ways : 

1. By grease cups, 

2. By oil cups. 

3. By oily pads of felt or waste. 

4. By oil wells with rings or chains for lifting the 
oil. 

5. By centrifugal force through a hole in the journal 
itself. 

Grease cups have little to recommend them except 



100 



MACHINE DESIGN. 



as auxiliary safety devices. Oil cups are various in 
their shapes and methods of operation and constitute 
the cheap class of lubricating devices. They may be 
divided according to their operation into wick oilers, 
needle feed, and sight feed. The two first mentioned 
are nearly obsolete and the sight feed oil cup, which 
drops the oil at regular intervals through a glass tube 
in plain sight, is in common use. The best sight feed 
oiler is that which can be readily adjusted as to time 
intervals, which can be turned on or off without dis- 
turbing the adjustment and which shows clearly by its 
appearance whether it is turned on. On engines and 
electric machinery which is in continuous use day and 
night, it is very important that the oiler itself should 
be stationary, so that it may be filled without stopping 
the machinery. 

A modern sight feed oiler for an engine is illustrated 

in Fig. 43. T is the glass tube where the oil drop 

is seen. The feed is regulated by the 

Q nut N, while the lever L shuts off the 

XX oil. Where the lever is as shown the 

JoT •» 

oil is dropping, when horizontal the 
oil is shut off. 

The nut can be adjusted once for 
all, and the position of the lever shows 
immediately whether or not the cup is 
in use. 

In modern engines particular at- 
tention has been paid to the problem 
/A7\\ of continuous oiling. The oil cups are 

all stationary and various ingenious 
devices are used for catching the drops 
of oil from the cups and distributing 
them to the bearing surfaces. 





Fig 43. 



LUBRICATION. 



101 



For continuous oiling of stationary bearings as in 
line shafting and electric machinery, an oil well below 
the bearing is preferred, with some automatic means 
of pumping the oil over the bearing, when it runs 
back by gravity into the well. Por- 
ous wicks and pads acting by capil- 
lary attraction are uncertain in 
their action and liable to become 
clogged. For bearings of medium 
size, one or more light steel rings 
running loose on the shaft and dip- 
ping into the oil, as shown in Fig. 44, 
are the best. For large bearings 
flexible chains are employed which 
take up less room than the .ring, 
are most used on parts which cannot readily be oiled 
when in motion, such as loose pulleys and the crank 
pins of engines. 

Fig. 45 shows two such devices as applied to an en- 
gine. In A the oil is supplied by the waste from the 
main journal ; in B an external sight-feed oil cup is 
used which supplies oil to the central revolving cup C. 




Fig. 44. 
Centrifugal oilers 




s. 



= ; rv 







c 



Fig. 45. 

Loose pulleys or pulleys running on stationary studs 
are best oiled from a hole running along the axis 



102 



MACHINE DESIGN. 



of the shaft and thence out radially to the surface of 
the bearing. See Fig 46. A loose bushing of some 
soft metal perforated with holes is a good safety device 
for loose pulleys. 




Fig. 46. 
Note : For adjustable pedestal and hanging bear- 
ings see the chapter on shafting. 

45: Friction of Journals : 
Let W=the total load of a journal in lbs. 
?=the length of journal in inches. 
d=the diameter of journal in inches. 
N= number of revolutions per minute. 
V— velocity of rubbing in feet per minute. 
F= friction at surface of journal in lbs. 
= W tan iff nearly, whose iff is the angle of re- 
pose for the two materials. 

If a journal is properly fitted in its bearing and does 
not bind, the value of F will not exceed W tan if/ and 
may be slightly less. The value of tan if/ varies accord- 
ing to the materials used and the kind of lubrication, 
from .05 to .01 or even less. See experiments described 
in Art. 48. The work absorbed in friction may be thus 
expressed : 
Fv= Wtan * ^dN = -dNWtar>4 ft lbg _ per min (57) 



LIMITS OF PRESSURE. 103 

46. Limits of Pressure. Too great an intensity of 
pressure between the surface of a journal and its bear- 
ing will force out the lubricant and cause heating 
and possibly " seizing." The safe limit of pressure de- 
pends on the kind of lubricant, the manner of its ap- 
plication and upon whether the pressure is continuous 
or intermittent. The projected area of a journal, or 
the product of its length by its diameter, is used as a 
divisor. 

The journals of railway cars offer a good example 
of continuous pressure and severe service. A limit of 
300 pounds per square inch of projected area has been 
generally adopted in such cases. 

In the crank and wrist pins of engines, the reversal 
of pressure diminishes the chances of the lubricant 
being squeezed out, and a pressure of 500 lbs. per 
sq. in. is generally allowed. 

The use of heavy oils or of an oil bath, and the em- 
ployment of harder materials for the journal and its 
bearing allow of even greater pressures. 

Professor Barr's investigations of steam engine pro- 
portions * show that the pressure per square inch on 
the cross-head pin varies from ten to twenty times that 
on the piston, while the intensity of pressure on the 
crank pin is from two to eight times that on the piston. 
Allowing a mean pressure on the piston of fifty pounds 
per square inch would give the following range of 
pressures : 

Minimum. Maximum. 
Wrist pins. . . 500 1000 

Crank pins. . . 100 400 

The larger values for the wrist pins are allowable on 

* Trans. A. S. M. E., Vol. XVIII. 






101 MACHINE DESIGN. 

account of the comparatively low velocity of rubbing. 
Naturally the larger values for the pressure are found 
in the low speed engines. 

A discussion of the subject of bearings is reported in 
the transactions of the American Society of Mechanical 
Engineers for 1905-06 * and some valuable data are 
furnished. Mr. Geo. M. Basford says that locomotive 
crank pins have been loaded as high as 1500 to 1700 
pounds per square inch, and wrist pins to 4000 pounds 
per square inch. 

Locomotive driving journals on the other hand are 
limited to the following pressures : 

Passenger locomotives . 190 lbs. per sq. in. 
Freight " . 200 

Switching " .220 

Cars and tender bearings. 300 

Mr. H. G. Keist gives some figures on the practice 
of the General Electric Company, for motors and 
generators. 

This company allows from 30 to 80 pounds pressure 
per square inch with an average value of from 40 to 45 
pounds. The rubbing speeds vary from 40 feet to 1200 
feet per minute. Mr. Eeist quotes approvingly the 
formula of Dr. Thurston's, viz : That the product of 
the pressure in pounds per square inch and the rubbing 
speed in feet per minute should not exceed 50,000. 

A careful reading of the whole discussion will repay 
any one who has to design shaft bearings of any de- 
scription. 

47. Heating of Journals. The proper length of 
journals depends on the liability of heating. 

* Trans. A. S. M. R, Vol. XXVII. 



HEATING OF JOURNALS. 105 

The energy or work expended in overcoming friction 
is converted into heat and must be conveyed away by 
the material of the rubbing surfaces. If the ratio of 
this energy to the area of the surface exceeds a certain 
limit, depending on circumstances, the heat will not 
be conveyed away with sufficient rapidity and the 
bearing will heat. 

The area of the rubbing surface is proportional to 
the projected area or product of the length and diame- 
ter of the journal, and it is this latter area which is 
used in calculation. 

Adopting the same notation as is used in Art. 45, 
we have from equation (57). 

the work of friction J^ dN ^ an * m ft. lbs. per min. 

or =7rclNWtan^ inch lbs. 

The work per square inch of projected area is then : 

irdNWtanilr irNWtanxp , , 

W= Id = I W 

Solving in (a) for I 

, TrNWtanxj/ /,s 

w • \ ) 

Let ™ , == C a co-efficient whose value is to be ob- 
Trtanif/ 

tained by experiment ; then 

C =I^andZ=I™ . . (58) 

Crank pins of steam engines have perhaps caused 
more trouble by heating than any other form of jour- 
nal. A comparison of eight different classes of propel- 
lers in the old U. S. Navy showed an average value 
for C of 350,000. 



106 MACHINE DESIGN. 

A similar average for the crank pins of thirteen 
screw steamers in the French Navy gave C= 400,000. 

Locomotive crank pins which are in rapid motion 
through the cool outside air allow a much larger value 
of C, sometimes more than a million. 

Examination of* ten modern stationary engines 
shows an average value of (7=200,000 and an average 
pressure per square inch of projected area =300 lb. 

The investigations of Professor Barr above referred 
to show a wide variation in the constants for the length 
of crank pins in stationary engines. He prefers to use 

HP 
the formula : l=K-j-+B where K and B are con- 
stants and L= length of stroke of engine in inches. 
We may put this in another form since : 

- r = 1ftQAnn where IF is the total mean pressure. 

The formula then becomes : 

l = K mm +B • • • • m 

The value of B was found to be 2.5 in. for high- 
speed and 2 in. for low-speed engines, while K fluc- 
tuated from .13 to .46 with an average of .30 in the 
the former class, and from .40 to .80 with an average 
of . 60 in the low-speed engines. 

If we adopt average values we have the following 
formulas for the crank-pins of modern stationary 
engines : 

High-speed engines 1= qqqqoq + ^-S ^ n - • . • -(60) 
Low-speed engines Z~= QOQQQn +2in (^-0 



EXPERIMENTS. 107 

Compare these formulas with (58) when values of C 
are introduced. 

In a discussion on the subject of journal bearings in 
1885,* Mr. Geo. H. Babcock said that he had found it 
practicable to allow as high as 1200 lb. per sq. in. on 
crank pins while the main journal could not carry 
over 300 lb. per sq. in. without heating. One rule 
for speed and pressure of journal bearings used by a 
well-known designer of Corliss engines is to multiply 
the square root of the speed in feet per second by the 
pressure per square inch of projected area and limit 
this product to 350 for horizontal engines and 500 in 
vertical engines. 

48. Experiments. Some tests made on a steel journal 
3J inches in diameter and 8 inches long running in a 
cast-iron bearing and lubricated by a sight-feed oiler, 
will serve to illustrate the friction and heating of such 
journals. 

The two halves of the bearing were forced together 
by helical springs with a total force of 1400 pounds, so 
that there was a pressure of 54 lb. per sq. in. on each 
half. The surface speed was 430 ft. per min. and the 
oil was fed at the rate of about 12 drops per minute. 
The lubricant used was a rather heavy automobile oil 
having a specific gravity of 0.925 and a viscosity of 
174 when compared with water at 20 deg. Cent. 

The length of the run was two hours and the tem- 
perature of the room 70 deg. Fahr. (See Table XXII.) 

* Trans. A. S. M. E., Vol. VI. 



108 



MACHINE DESIGN. 

TABLE XXII. 

FRICTION OF JOURNAL BEARING. 



Time. 


Rev. per min. 


Temp. Fahr. 


Coeff . of friction. 


10:03 


500 


69 


.024 


10:15 


482 


82 


.0175 


10:30 


506 


100 


.013 


10:45 


506 


115 


.010 


11:00 


516 


125 


.010 


11:15 




135 


.001 


11:30 




145 


.004 


11 : 45 


512 


147 


.004 


12:00 




151 


.007 



49. Strength and Stiffness of Journals. A journal 
is usually in the condition of a bracket with a uniform 

load, and the bending moment M=-~- 

Therefore by formula (6) 



cl 



-i 



10.21T 3|5.1W7 



s ~ 

or d=1.721 



1 



S 

m 

s ' 



(62) 



The maximum deflection of such a bracket is 

* wr 



1 



'SE I 



d 4 



64 ~8E& 

uwr 2.547 wr 



8ttE& ~ E& 
If as is usual A is allowed to be T \^ inches, then for 

stiffness .(63) 

(64) 



4 I WP 
or approximately d=4.J— ^— . 



CAPS AND BOLTS. 109 

The designer must be guided by circumstances in 
determining whether the journal shall be calculated 
for wear, for strength or for stiffness. A safe way is 
to design the journal by the formulas for heating and 
wear and then to test for strength and deflection. 

Eemember that no factor of safety is needed in 
formula for stiffness. 

Note that W in formulas for strength and stiffness 
is not the average but the maximum load. 

50. Caps and Bolts. The cap of a journal bearing 
exposed to upward pressure is in the condition of a 
beam supported by the holding down bolts and loaded 
at the center, and may be designed either for strength 
or for stiffness. 
Let : P=max. upward pressure on cap. 

L= distance between bolts. 

b= breadth of cap at center. 

h= depth of cap at center. 

A — greatest allowable deflection. 

Strength : M=^*=^ 



=VS (65 > 



Stiffness : 



h 

A 



4:SEI 



T bh z WU 

12 ~4AE± 



^ 



{m) 



46£'A 

If A is allowed to be T -^ inches and E for cast iron is 
taken =18000000 

then: h^.OlWL^j—- (67) 



HO MACHINE DESIGN. 

The holding down bolts should be so designed that 
the bolts on one side of the cap may be capable of 
carrying safely two thirds of the total pressure. 

Problems. 

1 . A flat car weighs 10 tons, is designed to carry a load of 
20 tons more and is supported by two four-wheeled trucks, the 
axle journals being of wrought iron and the wheels 33 inches 
in diameter. 

Design the journals, considering heating, wear, strength 
and stiffness, assuming a maximum speed of 30 miles an hour, 
factor of safety =10 and C= 300000. 

2. The following dimensions are those generally used for the 
journals of freight cars having nominal capacities as indicated : 



CAPACITY. 


DIMENSIONS OF JOURNAL, 


100000 lb. 


4.5 by 9 in. 


60000 lb. 


4.25 by 8 in. 


40000 lb. 


3.75 by 7 in. 



Assuming the weight of the car to be 40 per cent of its 
carrying capacity in each instance, determine the pressure per 
square inch of projected area and the value of the constant C 
{Formula (58)} . 

3. Measure the crank pin of any modern engine which is 
accessible, calculate the various constants and compare them 
with those given in this chapter. 

4. Design a crank pin for an engine under the following con- 
ditions : 

Diameter of piston =28 inches. 

Maximum steam pressure=90 lb. per sq. in. 
Mean steam pressure =40 lb. per sq. in. 

Revolutions per minute =75 
Determine dimensions necessary to prevent wear and heating 
and then test for strength and stiffness. 

5. Design a crank pin for a high speed engine having the 
following dimensions and conditions : 

Diameter of piston = 14 inches 



STEP-BEARINGS. HI 

Maximum steam pressure=100 lb. per sq. in. 
Mean steam pressure =50 lb. per sq. in. 

Revolutions per minute=250. 

6. Make a careful study and sketch of journals and journal 
bearings on each of the following machines and analyze as to 
(a) Materials, (b) Adjustment, (c) Lubrication. 

a. An engine lathe. 

b. A milling machine. 

c. A steam engine. 

d. An electric generator or motor. 

7. Sketch at least two forms of oil cup used in the labora- 
tories and explain their working. 

8. The shaft journal of a vertical engine is 4 in. in diameter 
by 6 in. long. The cap is of cast iron, held down by 4 bolts 
of wrought iron, each 5 in. from center of shaft, and the 
greatest vertical pressure is 12000 lb. 

Calculate depth of cap at center for both strength and stiff- 
ness, and also the diameter of bolts. 

9. Investigate the strength of the cap and bolts of some 
pillow block whose dimensions are known, under a pressure of 
500 lb. per sq. in. of projected area. 

10. The total weight on the drivers of a locomotive is 
64000 lb. The drivers are four in number, 5 ft. 2 in. in 
diameter, and have journals 7| in. in diameter. 

Determine horse power consumed in friction when the 
locomotive is running 50 miles an hour, assuming tan0=.O5. 

51. Step-Bearings. Any bearing which is designed 
to resist end thrust of the shaft rather than lateral 
pressure is denominated a step or thrust bearing. 
These are naturally most used on vertical shafts, but 
may be frequently seen on horizontal ones as for ex- 
ample on the spindles of engine lathes, boring machines 
and milling machines. 

Step-bearings may be classified according to the 
shape of the rubbing surface, as flat pivots and collars, 
conical pivots, and conoidal pivots of which the Schjele 



112 MACHINE DESIGN. 

pivot is the best known. When a step-bearing on a 
vertical shaft is exposed to great pressure or speed it 
is sometimes lubricated by an oil tube coming up from 
below to the center of the bearing and connecting with 
a stand pipe or force-pump. The oil entering at the 
center is distributed by centrifugal force* 

52. Friction of Pivots or Step-bearings. — Flat 
Pivots. 

Let W— weight on pivot 

di=: outer diameter of pivot 

p= intensity of vertical pressure 

T= moment of friction 

f= co-efficient of friction =tan <f> 

We will assume p to be a constant which is no doubt 
approximately true, 

* area ird\ 

Let r=the radius of any elementary ring of a width 
= dr y then area of element =2?r rdr 

Friction of element =fpX < 2'n-rdr 
Moment of friction of element = 2fjJTrr 2 dr 



and T 



=2fp*f^r*dr (a) 

or T=2fp^=2fp*^ 

= 2T X ^f = 3 Wfd >* ' (68) 

The great objection to this form of pivot is the un- 
even wear due to the difference in velocity between 
center and circumference. 



CONICAL PIVOT. 113 

53. Flat Collar. 

Let d 2 =inside diameter 
Integrating as in equation (a) above, but using 

limits ~ -and-^ 2 we have 

In this case 

P -Tr{d\-dt) 

and T =\Wf^f=^ (69) 

54. Conical Pivot. 

Let a= angle of inclination to the vertical. 

I c — d i A As in the case of a flat 

\ j J / ring the intensity of the 

\ I J \\J vertical pressure is 

\ / and the vertical pressure 

X^ / on an elementary ring of 

\j/ the bearing surface is 

Fig. 47. 

dW = -** ^ r JWr& 

Tr(dl—di) d\—d\ 

As seen in Fig. 47 the normal pressure on the 
elementary ring is 

dp= dW = SWrdr 



sina (d'i—d 2 o)sina 
8 



11-t MACHINE DESIGN. 

The friction on the ring is/cLPand the moment of 
this friction is 

dT-frdP-Mg*- 

J (d\—dl)sina 



\d\—dl)sinaj ck 



*"£—£«* 



2 

_iWf ti-® (m 

As a approaches ~ the value of T approaches that of a 

flat ring, and as a approaches the value of T ap- 
proaches go . 
If dj=0 we have 

T=$W&- (71) 

The conical pivot also wears unevenly, usually as- 
suming a concave shape as seen in profile. 

55. Schiele's Pivot. By experimenting with a 
pivot and bearing made of some friable material, it was 
shown that the outline tended to become curved as 
shown in Fig. 49. This led to a mathematical investi- 
gation which showed that the curve would be a trac- 
trix under certain conditions. 

This curve may be traced me- 
, 5 C^\ chanically as shown in Fig. 48. 

W Let the weight W be free to 

move on a plane. Let the string 
SW be kept taut and the end S 
moved along the straight line SL. 
Then will a pencil attached to the 

TO AO center of W trace on the plane a 

Fig. 48; . r 

tractrix whose axis is SL, 



SCHIELES PIVOT. 



115 



In Fig. 49 let SW= length 
of string =r\ and let P be any 
point in the curve. Then it is 
evident that the tangent PQ to 
the curve is a constant and =?\ 

Also 



Let a pivot be generated by 
revolving the curve around its 
axis SL. As in the case of the 
conical pivot it can be proved 
that the normal pressure on 
an element of convex surface is 




Fig. 49. 



dP = 



SWrdr 



(a) 



(dl—d'$)sind' 

Let the normal wear of the pivot be assumed to be 
proportional to this normal pressure and to the velocity 
of the rubbing surfaces, i. e. normal wear proportional 
to pr, then is the vertical wear proportional to 



pr 
sinB' 



But 



is a constant, therefore the vertical 



sind 

wear will be the same at all points. This is the 
characteristic feature and advantage of this form of 
pivot. 

As shown in equation (a) 
SW^dr 

Cli — CL% 



dP 



rjrp SWfr.rdr 



di 



d\ 



and 



T _%Wfr 1 r\-r\ Wfd x 
1 ~ dl-dl 2 "~~ 2 ' 



.(72) 



T is thus shown to be independent of d, or of the* 
length of pivot used, 



116 MACHINE DESIGN. 

This pivot is sometimes wrongly called antifriction. 
As will be seen by comparing equations (68) and (72) 
the moment of friction is fifty per cent, greater than 
that of the common flat pivot. 

The distinct advantage of the Schiele pivot is in the 
fact that it maintains its shape as it wears and is self- 
adjusting. It is an expensive bearing to manufacture 
and is seldom used on that account. 

It is not suitable for a bearing where most of the 
pressure is side ways. 

56. Multiple Bearings. To guard against abrasion 
in flat pivots a series of rubbing surfaces which divide 
the wear is sometimes provided. Several flat discs 
placed beneath the pivot and turning indifferently, 
may be used. Sometimes the discs are made alter- 
nately of a hard and a soft material. Bronze, steel 
and raw hide are the more common materials. 

Notice in this connection the button or washer at 
the outer end of the head spindle of an engine lathe 
and the loose collar on the main journal of a milling 
machine. See Figs. 41 and 42. Pivots are usually 
lubricated through a hole at the center of the bearing 
and it is desirable to have a pressure head on the oil to 
force it in. 

The compound thrust bearing generally used for 
propeller shafts consists of a number of collars of the 
same size forged on the shafts at regular intervals and 
dividing the end thrust between them, thus reducing 
the intensity of pressure to a safe limit without making 
the collars unreasonably large. 

A safe value for p the intensity of pressure is, ac- 
cording to Whitham, 60 lb. per sq. in. for high speed 



MULTIPLE BEARINGS. 117 

A table given by Prof. Jones in his book on Machine 
Design shows the practice at the Newport News ship- 
yards on marine engines of from 250 to 5000 H. P. 
The outer diameter of collars is about one and one-half 
times the diameter of the shafts in each case and the 
number of collars used varies from 6 in the smallest 
engine to 11 in the largest. The pressure per sq. in. of 
bearing surface varies from 18 to 46 lb. with an 
average value of about 32 lb. 

The hydraulic foot step sometimes used for the 
vertical shafts of turbines is in effect a rotating plunger 
supported by water pressure underneath and so packed 
in its bearing as to allow a slight leakage of water for 
cooling and lubricating the bearing surfaces. 

Problems. 

1. Design and draw to full size a Schiele pivot for a water 
wheel shaft 4 inches in diameter, the total length of the bear- 
ing being 3 inches. 

Calculate the horse-power expended in friction if the total 
vertical pressure on the pivot is two tons and the wheel makes 
150 revs, per min. and assuming /=. 25 for metal on wet wood. 

2. Compare the friction of the pivot in Prob. 1, with that of 
a flat collar of the same projected area and also with that 
of a conical pivot having a =30 deg. 

3. Design a compound thrust bearing for a propeller shaft 
the diameters being 14 and 21 inches, the total thrust being 
80,000 lbs. and the pressure 40 lb. per sq. in. 

Calculate the horse-power consumed in friction and compare 
with that developed if a single collar of same area had been 
used. Assume /=. 05 and rev. per min. =120. 



CHAPTER VIII. 

BALL AND ROLLER BEARINGS. 

57. General Principles. The object of interposing a 
ball or roller between a journal and its bearing, is to 
substitute rolling for sliding friction and thus to re- 
duce the resistance. This can be done only partially 
and by the observance of certain principles. In the 
first place it must be remembered that each ball can 
roll about but one axis at a time ; that axis must be 
determined and the points of contact located accord- 
ingly. 

Secondly, the pressure should be approximately 
normal to the surfaces at the points of contact 

Finally it must be understood, that on account of 
the contact surfaces being so minute, a comparative!}" 
slight pressure will cause distortion of the balls and an 
entire change in the conditions. 

58. Journal Bearings. These may be either two, 
three or four point, so named from the number of 
points of contact of each ball. 

The axis of the ball may be assumed as parallel or 
inclined to the axis of the journal and the points of 
contact arranged accordingly. The simplest form con- 
sists of a plain cylindrical journal running in a bearing 
of the same shape and having rings of balls interposed. 
The successive rings of balls should be separated by 
thin loose collars to keep them in place. These collars 
are a source of rubbing friction, and to do away with 
them the balls are sometimes run in grooves either in 
journal, bearing or both. 
118 



JOURNAL BEARINGS. 



119 




Fig. 50 shows a bearing of this type, there being 
three points of contact and the axis of ball being 
parallel to that of journal. 

The bearings so far 
mentioned have no 
means of adjustment 
for wear. Conical 
bearings, or those in 
which the axes of the 
balls meet in a com- 
mon point, supply 
this deficiency. In designing this class of bearings, 
either for side or end thrust, the inclination of the 
axis is assumed according to the obliquity desired and 
the points of contact are then so located that there 
shall be no slipping. 

Fig. 51 illustrates a common form of adjustable or 
cone bearing and shows the method of designing a 
three point contact. A C 
is the axis of the cone, 
while the shaded area is a 
section of the cup, so 
called. Let a and b be 
two points of contact be- 
tween ball and cup. Draw 
the line a b and produce 
to cut axis in A. Through 
the center of ball draw 
the line A B ; then will 
this be the axis of rotation of the ball and a c,bd 
will be the projections of two circles of rotation. As 
the radii of these circles have the same ratio as the radii 
of revolution a n, b m, there will be no slipping and 
the ball will roll as a cone inside another cone. The 




120 



MACHINE DESIGN. 



exact location of the third point of contact is not 
material. If it were at c, too much pressure would 
come on the cup at b ; if at d there would be an excess 
of pressure at a, hut the rolling would be correct in 
either case. A convenient method is to locate p by 
drawing A D tangent to ball circle as shown. It is 
recommended however that the two opposing surfaces 
at p and & or a should make with each other an angle 
of not less than 25 deg. to avoid sticking of the ball. 

To convert the bearing just shown to four point 
contact, it would only be necessary to change the one 
cone into two cones tangent to the ball at c and d. 

To reduce it to two point contact the points a and b 
are brought together to a point opposite p. As in this 
last case the ball would not be confined to a definite 
path it is customary to make one or both surfaces con- 
cave conoids with a radius about three fourths the 
diameter of the ball. See Fig. 52. 

59. Step-Bearings. The 
same principles apply as 
in the preceding article 
and the axis and points 
of contact may be varied 
in the same way. The 
most common form of 
step-bearing consists of 
two flat circular plates 
Fig. 52. separated by one or more 

rings of balls. Each ring must be kept in place by 
one or more loose retaining collars, and these in turn 
are the cause of some sliding friction. This is a bear- 
ing with two point contact and the balls turning on 
horizontal axes. If the space between the plates is 
filled with loose balls, as is sometimes done, the rubbing 




STEP-BEARINGS. 



121 



of the balls against each other will cause considerable 
friction. 

To guide the balls without rubbing friction three 
point contact is generally used. 

Fig. 53 illustrates a bearing of this character. The 



a b 



■> 




k- 


1 p 


i 


B d (\r 


\A n 




Wmm, 


ID 


mi 





Fig. 53. 



method of design is shown in the figure, the principle 
being the same as in Fig. 51. By comparing the letter- 
ing of the two figures the similarity will be readily seen. 

This last bearing may 
be converted to four point 
contact by making the upper 
collar of the same shape as 
the lower. To guide the 
balls in two point contact 
use is sometimes made of 
a cage ring, a flat collar 
drilled with holes just a 
trifle larger than the balls 
and disposing them either 
in spirals or in irregular Fig. 54. 

order. See Fig. 54. 

This method has 




the advantage 



of making each 



122 MACHINE DESIGN. 

ball move in a path of different radius thus securing 
more even wear for the plates. 

60. Materials and Wear. The balls themselves are 
always made of steel, hardened in oil, tempered and 
ground. They are usually accurate to within one ten 
thousandth of an inch. The plates, rings and journals 
must be hardened and ground in the same way and 
perhaps are more likely to wear out or fail than the 
balls. A long series of experiments made at the Case 
School of Applied Science on the friction and endur- 
ance of ball step-bearings showed some interesting 
peculiarities. 

Using flat plates with one circle of quarter inch balls 
it was found that the balls pressed outward on the 
retaining ring with such force as to cut and indent it 
seriously. This was probably due to the fact that the 
pressure slightly distorted the balls and changed each 
sphere into a partial cylinder at the touching points. 
While of this shape it would tend to roll in a straight 
line or a tangent to the circle. Grinding the plates 
slightly convex at an angle of one to one and-a-half 
degrees obviated the difficulty to a certain extent. 
Under even moderately heavy loads the continued 
rolling of the ring of balls in one path soon damaged 
the plates to such an extent as to ruin the bearing. 

A flat bearing filled with loose balls developed three 
or four times the friction of the single ring and a three 
point bearing similar to that in Fig. 53 showed more 
than twice the friction of the two point. 

A flat ring cage such as has already been described 
was the most satisfactory as regards friction and en- 
durance. 
' The general conclusions derived from the experi- 



ROLLER BEARINGS. 123 

merits were that under comparatively light pressures 
the balls are distorted sufficiently to seriously disturb 
the manner of rolling and that it is the elasticity and 
not the compressive strength of the balls which must 
be considered in designing bearings. 

61. Design of Bearings. Figures on the direct 
crushing strength of steel balls have little value for the 
designer. For instance it has been proved by numer- 
ous tests that the average crushing strengths of i inch 
and | inch balls are about 7500 lb. and 15000 lb. 
respectively. Experiments made by the writer show 
that a \ inch ball loses all value as a transmission 
element on account of distortion, at any load of more 
than 100 lb. 

Prof. Gray states, as a conclusion from some ex- 
periments made by him, that not more than 40 lb. per 
ball should be allowed for f inch balls. 

This distortion doubtless accounts for the failure of 
theoretically correct bearings to behave as was ex- 
pected of them. Ball bearings should be designed as 
has been explained in the preceding articles and then 
only used for light loads. 

62. Roller Bearings. The principal disadvantage 
of ball bearings lies in the fact that contact is only at 
a point and that even moderate pressure causes exces- 
sive distortion and wear. The substitution of cylinders 
or cones for the balls is intended to overcome this 
difficulty. 

The simplest form of roller bearing consists of a 
plain cylindrical journal and bearing with small cylin- 
drical rollers interposed instead of balls. There are 
two difficulties here to be overcome. The rollers tend 



124 



MACHINE DESIGN. 



to work endways and rub or score whatever retains 
them. They also tend to twist around and become 

unevenly worn or even 



bent and broken, unless 
held in place by some sort 
of cage. In short they 
will not work properly 
unless guided and any 
form of guide entails 



nn 



Fig. 55. 



B 



) 



sliding friction. The cage generally used is a cylin- 
drical sleeve having longitudinal slots which hold 
the rollers loosely and prevent their getting out of 
place either sideways or endways. 

The use of balls or convex washers at the ends of the 
rollers has been tried with some degree of success. 
See Fig. 55. Large rollers have been turned smaller 
at the ends and the bearings then formed allowed to 
turn in holes bored in revolving collars. These collars 
must be so fastened or - geared together as to turn in 
unison. 

63. Grant Roller Bearing. The Grant roller is conical 
and forms an intermediate between the ball and the 
cylindrical roller having some 
of the advantages of each. 
The principle is much the 
same as in the adjustable ball 
bearing, Fig. 52, rolling cones 
being substituted for balls, 
Fig. 56. The inner cone turns 
loose on the spindle. The 
conical rollers are held in posi- 
tion by rings at each end, while the outer or hollow 
cone ring is adjustable along the axis. 




Fig. 56. 



HYATT ROLLERS. 125 

Two sets of cones are used on a bearing, one at each 
end to neutralize the end thrust, the same as with ball 
bearings. 

64. Hyatt Rollers. The tendency of the rollers to 
get out of alignment has been already noticed. The 
Hyatt roller is intended by its flexibility to secure 
uniform pressure and wear under such conditions. It 
consists of a flat strip of steel wound spirally about 
a mandrel so as to form a continuous hollow cylinder. 
It is true in form and comparatively rigid against com- 
pression, but possesses sufficient flexibility to adapt 
itself to slight changes of bearing surface. 

Experiments made by the Franklin Institute show 
that the Hyatt roller possesses a great advantage in 
efficiency over the solid roller. 

Testing § inch rollers between flat plates under loads 
increasing to 550 lb. per linear inch of roller developed 
co-efficients of friction for the Hyatt roller from 23 to 
51 per cent, less than for the solid roller. Subsequent 
examination of the plates showed also a much more 
even distribution of pressure for the former. 

A series of tests were conducted by the writer in 
1904—05 to determine the relative efficiency of roller 
bearings, as compared with plain cast iron and bab- 
bitted bearings under similar conditions. 1 The bear- 
ings tested had diameters of Iff, 2^, 2^, and 2-Jf 
inches and lengths approximately four times the 
diameters. In the first set of experiments Hyatt roller 
bearings were compared with plain cast iron sleeves, 
at a uniform speed of 480 rev. per min. and under loads 
varying from 64 to 264 pounds. The cast iron bearings 
were copiously oiled. 

1 Machinery. N. Y., Oct. 1905. 



126 



MACHINE DESIGN. 



As the load was gradually increased, the value of / 
the coefficient of friction remained nearly constant 
with the plain bearings, but gradually decreased in the 
case of the roller bearings. Table XXIII. gives a 
summary of this series of tests. 

TABLE XXIII. 

COEFFICIENTS OF FRICTION FOR ROLLER AND PLAIN 

BEARINGS. 



Diameter 

of 
Journal. 



Hyatt Bearing 


Plain Bear 


mg. 


Max. 


Min. j Ave. 


Max, 


Min. 


Ave. 


.036 


.019 .026 


.160 


.099 


.117 


.052 


.034 .040 


.129 


.071 


.094 


.041 


.025 .030 


.143 


.076 


.104 


.053 


.049 .051 


.138 


.091 


.104 



The relatively high value of / in the 2 T 3 g- and 2ff 
roller bearings were due to the snugness of the fit be- 
tween the journal and the bearing, and show the ad- 
visability of an easy fit as in ordinary bearings. 

The same Hyatt bearings were used in the second 
set of experiments, but were compared with the 
McKeel solid roller bearings and with plain babbited 
bearings freely oiled. The McKeel bearings contained 
rolls turned from solid steel and guided by spherical 
ends fitting recesses in cage rings at each end. The 
cage rings were joined to each other by steel rods par- 
allel to the rolls. The journals were run at a speed of 
560 rev. per min. and under loads varying from 113 to 
456 pounds. Table XXIV. gives a summary of the 
second series of tests. 



ROLLER STEP-BEARINGS. 



127 



TABLE XXIV. 

COEFFICIENTS OF FRICTION FOR ROLLER AND PLAIN 
BEARINGS. 



Diam. 

of 
Jo'rnal . 


Hyatt Bearing. 


McKeel Bearing. 


Babbitt bearing. 


Max. 


Min. 


Ave. 


Max. 


Min. 


Ave. 


Max. 


Min. 


Ave. 


2-4- 


.032 
.019 
.042 
.029 


-.012 
.011 
.025 
.022 


.018 
.014 
.032 
.025 


.033 


.017 


.022 


.074 
.088 
.114 
.125 


.029 

.078 
.083 
.089 


.043 

.082 




.028 
.039 


.015 
.019 


.021 

.027 


.096 

.107 



The variation in the values for the babbited bearing 
is due to the changes in the quantity and temperature 
of the oil. For heavy pressures it is probable that the 
plain bearing might be more serviceable than the others. 
Notice the low values for /in Table XXII. 

Under a load of 470 pounds the Hyatt bearing de- 
veloped an end thrust of 13.5 pounds and the McKeel 
one of 11 pounds. 

This is due to a slight skewing of the rolls and 
varies, sometimes reversing in direction. 

If roller bearings are properly adjusted and not 
overloaded a saving of from two-thirds to three-fourths 
of the friction may be reasonably expected. 

65. Roller Step-Bearings. In article 60 attention 
was called to the fact that the balls in a step-bearing 
under moderately heavy pressures tend to become 
cylinders or cones and to roll accordingly. This has 
suggested the use of small cones in place of the balls, 
rolling between plates one or both of which are also con- 
ical, A successful bearing of this kind with short 



128 



MACHINE DESIGN. 



cylinders in place of cones is used by the Sprague-Pratt 
Elevator Co., and is described in the American Machin- 
ist for June 27, 1901. The rollers are arranged in two 
spiral rows so as to distribute the wear evenly over the 
plates and are held loosely in a flat ring cage. This 
bearing has run well in practice under loads double 
those allowable for ball bearings, or over 100 lb. per 
roll for rolls one-half inch in diameter and one-quarter 
inch long. 

Fig. 57 illustrates a bearing of this character. Col- 
lars similar to this have been used in thrust bearings 

for propeller shafts. The 
discussion referred to in 
Art. 46 also included ball 
and roller bearings and 
should be read by the de- 
signer. Mr. Mossberg, de- 
signer of the roller bear- 
ings of that name, recom- 
mends rollers of spring 
tempered tool steel, cages 
of tough bronze and boxes 
of high carbon steel with a 
hard temper. Mr. Charles 
R. Pratt reports the limit of work for | inch balls in 
thrust bearings to be 100 pounds per ball at 700 revolu- 
tions per minute and 6 inches diameter circle of rotation. 
Mr. W. S. Rogers gives the maximum load for a 
1 inch ball as 1000 pounds and for a \ inch ball as 200 
pounds. Mr. Henry Hess states that in a roller bearr 
ing one fifth of the number of rollers multiplied by the 
length and diameter of one roller may be considered as 
the projected area of the journal. For ball bearings 
one fifth the total number of balls multiplied by the 




Fig. 57. 



ROLLER STEP-BEARINGS. 129 

square of the ball diameter may be used in the same 
way. 

Space forbids reference to all of the many varieties 
of ball and roller bearings shown in manufacturers' 
catalogues. These are all subject to the laws and limit- 
ations mentioned in this chapter, 

While such bearings will be used more and more in 
the future, it must be understood that extremely high 
speeds or heavy pressures are unfavorable and in most 
cases prohibitive. 

Furthermore, unless a bearing of this character is 
carefully designed and well constructed it will prove 
to be worse than useless. 



CHAPTER IX. 

SHAFTING, COUPLINGS AND HANGERS. 

66. Strength of Shafting. 

Let D= diameter of the driving pulley or gear. 
N= number rev. per minute. 
P== force applied at rim. 
T= twisting moment. 

The distance through which P acts in one minute is 

irDN inches and work=P7rZ)iV"in. lb. per min. 

PD 
But -^— =T the moment, and 27riV=the angular 

velocity. 

.*. work = moment X angular velocity. 
One horse power= 33000 ft. lb. per min. 

= 396000 in. lb. per min. 

PttDN ZttTN 



HP= 



396000 ~ 396000 



HP =Sm ■'-•; •■• * (73) 

also r= 63026HP (W) 

p= 126050HP (75) 

The general formula for a circular shaft exposed to 
torsion alone is 



S 
130 



STRENGTH OF SHAFTING. 131 



But r= 630|SP by(74) 

where iV=no. rev. per min. 
Substituting in formula for d 



\ — ~sn — nearl y • • • C? 6 ) 

S may be given the following values : 

45000 for common turned shafting. 
50000 for cold rolled iron or soft steel. 
65000 for machinery steel. 
It is customary to use factors of safety for shafting 
as follows : 

Headshafts or prime movers 15 
Line shafting 10 

Short counters 6 

The large factor of safety for head shafts is used not 
only on account of the severe service to which such 
shafts are exposed, but also on account of the incon- 
venience and expense attendant on failure of so im- 
portant a part of the machinery. The factor of safety 
for line shafting is supposed to be large enough to 
allow for the transverse stresses produced by weight 
of pulleys, pull of belts, etc., since it is impracticable 
to calculate these accurately in most cases. 

Substituting the values of S and introducing factors 
of safety, we have the following formulas for the safe 
diameters of the various kinds of shafts. 



132 



MACHINE DESIGN. 



TABLE XXV. 
DIAMETERS OF SHAFTING. 



Kind of 


Material. 


Shaft. 


Com'n Iron 


Soft Steel 


Mach'y Steel 


Head Shaft. 

Line Shaft. 

Counter Shaft. 


8.50^ 


,5S^ 
4.00^ 
3.38^ 


4.20 *ia^ 

\ iV 
3.67 ^f 

3.10^ 



The Allis-Chalmers Co. base their tables for the 
horse power of wrought iron or mild steel shafting on 
the formula HP=cd*N where c has the following 
values : 

c 
Heavy or main shafting .008 

Shaft carrying gears .010 

Light shafting with pulleys .013 
This is equivalent to using values of S as 2570 lb., 
3200 lb. and 4170 lb. persq. in. in the respective classes 
— and would gi^e for co-efficients in Table XXV. the 
numbers 5, 4.64 and 4.25 which are somewhat larger 
than those given for similar cases in the table. 

A table published by Win. Sellers & Co. in their 
shafting catalogue — gives the horse powers of iron and 
steel shafts for given diameters and speeds. An investi- 



COUPLINGS. 



133 



gation of the table shows it to be based upon a value 
of about 4000 lb. for S or a co-efficient of 4.31 in Table 
XXV. 

In case there is a known bending moment M, com- 
bined with a known twisting moment T, then a re- 
sultant twisting moment 

is to be substituted for T in the formulas (73) to (75). 

Mr. J. B. Francis has published a table in the Journal 
of the Franklin Institute which gives the greatest 
admissible distance between bearings for line shafts of 
different diameters, when subject to no transverse 
forces except from their own weight. This distance 
varies from 16 feet for 2 inch shafts up to 26 feet for 
9 inch shafts, the span being proportional to the cube 
root of the diameter. The distance should be much 
less when the -shaft carries numerous pulleys with 
their belts. 

67. Couplings. The flange or plate coupling is most 
commonly used for fastening together adjacent lengths 
of shafting. 

Fig. 58 shows the pro- &%& 

portions of such a coup- 
ling. The flanges are 
turned accurately on all 
sides, are keyed to the 
shafts and the two are 
centered by the projec- 
tion of the shaft from 
one part into the other Fig 58# 

as shown at A. The 

bolts are turned to fit the holes loosely so as not to in- 
terfere with the alignment. 




134 



MACHINE DESIGN. 



The projecting rim as at B prevents danger from 
belts catching on the heads and nuts of the holts. 

The faces of this coupling should be trued up in a 
lathe after being keyed to the shaft. 

Jones and Laughlins in their shafting catalogue give 
the following proportions for flange couplings. 



Diam. of Shaft. 


Diam. of Hub. 


Length of Hub. 


Diam. of 
Coupling. 


2 


^ 


3i 


8 


n 


5f 


4| 


10 


3 


6f 


5i 


12 


H 


8 


H 


14 


4 


9 


7 


16 


5 


Hi 


8| 


20 



There are five bolts in each coupling. 
The sleeve coupling is neater in appearance than the 
flange coupling but is more complicated and expensive. 
In Fig. 59 is illustrated a neat and effective coupling 

5 




of this type. It consists of the sleeve S bored with 
two tapers and two threaded ends as shown. The two 
conical, split bushings B B are prevented from turning 
by the feather key K and are forced into the conical 
recesses by the two threaded collars C C and thereby 



CLUTCHES. 



135 



clamped firmly to the shaft. The key K also nicks 
slightly the center of the main sleeve S, thus locking 
the whole combination. 

Couplings similar to this have been in use in the 
Union Steel Screw Works, Cleveland, Ohio, for many 
years and have given good satisfaction. 

The Sellers coupling is of the type illustrated in Fig. 
59, but is tightened by three bolts running parallel to 
the shaft and taking the place of the collars C C. 

In another form of sleeve 
coupling the sleeve is split 
and clamped to the shaft by 
bolts passing through the 
two halves as illustrated in 
Fig. 60. 

The "muff" coupling, as 
its name implies is a plain 
sleeve slipped over the shafts 

at the point of junction, accurately fitted and held by 
a key running from end to end. It may be regarded 
as a permanent coupling since it is not readily removed. 



i 










i 


rAi 


1 ( 


rfh 


L 








i 


MJi 


L 

.J L 


iyj 





Fie. 60. 



68. Clutches. By the term clutch, is meant a coup- 
ling which may be readily disengaged so as to stop the 
following shaft or pulley. Clutch couplings are of 
two kinds, positive or jaw clutches and friction 
clutches. 

The jaw clutch consists of two hubs having sector 
shaped projections on the adjacent faces which may 
interlock. One of the couplings can be slid on its shaft 
to and from the other by means of a loose collar and 
yoke, so as to engage or disengage with its mate. 
This clutch has the serious disadvantage of not being 
readily engaged when either shaft is in motion. 



136 



MACHINE DESIGN. 



n 





Friction clutches are not so positive in action but can 
be engaged without difficulty and without stopping 
the driver. 

Three different classes of friction clutches may be 
distinguished according as the engaging members are 
flat rings, cones or cylinders. 

The Weston clutch, 
Fig. 61, belongs to the 
first named class. A 
series of rings inside a 
sleeve on the follower 
B interlocks with a 
similar series outside a 
smaller sleeve on the 
driver A somewhat as 
in a thrust bearing 
(Art. 56). Each ring 
can slide on its sleeve but must rotate with it. 

When the parts A and B are forced together the 
rings close up and engage by pairs, producing a 
considerable turning moment with a moderate end 
pressure. Let : 

P = pressure along axis. 
n= number of pairs of surfaces in contact. 
/ = coefficient of friction. 
r=mean radius of ring. 
T= turning moment. 
Then will : 

T=Pfnr (77) 

If the rings are alternately wood and iron, as is 
usually the case, /will have values ranging from 0.25 
to 0.50. 

The cone clutch consists of two conical frustra, one 
external and one internal, engaging one another and 



CLUTCHES. 



137 



driving by friction. Using the same notation as before, 
and letting a= angle between element of cone and 
axis, the normal pressure between the two surfaces 

P Pf 

—. — ■ and the friction will be : ■ . ' . 
sin a SI1I a 



will be 



Therefore : 



sin a 



(78) 



a should slightly exceed 5 deg. to prevent sticking 
and /will be at least 0.10 for dry iron on iron. 

Substituting / = 0.10 and sin a =0.125 we have 
T = 0.SPr as a convenient rule in designing. 

Fig. 62 illustrates the type of clutch more generally 
used on shafting for 
transmitting mod- 
erate quantities of 
power. 

As shown in the 
figure one member 
is attached to a loose 
pulley on the shaft, 
but this same type 
can be used for con- 
necting two in- 
dependent shafts. 

The ring or hoop 
H, finished inside 
and out, is gripped 
at intervals by pairs 
of jaws J J having wooden faces. 

These jaws are actuated as shown by toggles and 
levers connected with the slip ring R. The toggles 
are so adjusted as to pass by the center and lock in the 
gripping position. 

These clutches are convenient and durable but 




Fig. 62. 



138 



MACHINE DESIGN. 



occupy considerable room in proportion to their trans- 
mitting power. The Weston clutch is preferable for 
heavy loads. 

The roller clutch is much used on automatic ma- 
chinery as it combines the advantages of positive 
driving and friction engagement. A cylinder on the 
follower is embraced by a rotating ring carried by the 
driver. 

The ring has a number of recesses on its inner surface 
which hold hardened steel rollers. These recesses being 
deeper at one end allow the rollers to turn freely as 
long as they remain in the deep portions. 

The bottom of the recess is inclined to the tangent of 
the circle at an angle of from 9 to 14 deg. 

When by suitable mechanism the rollers are shifted 
to the shallow portions of the recesses they are im- 
mediately gripped between the ring and the cylinder 
and set the latter in motion. 

A clutch of this type is almost instantaneous in its 
action and is very powerful, being limited only by the 
strength of the materials of which it is composed. 

Several small rolls of different materials and diam- 
eters were tested by the writer in 1905 with the 
following results : 



Material. 


Diameter. 


Length. 


Set load. 


Ultimate load, 


Cast Iron 


0.375 


1.5 


5500 


12400 


c < 


0.75 


1.5 


6800 


19500 


n 


1.125 


1.5 


7800 


29700 


" 


0.4375 


1.5 


8800 


20000 


Soft Steel 


0.4375 


1.5 


11100 




Hard Steel 


0.4375 


1.5 


35000 





69. Coupling Bolts. The bolts used in the ordinary 
flange couplings are exposed to shearing, and their 



SHAFTING KEYS. 139 

combined shearing moment should equal the twisting 
moment on the shaft. 

Let 11 = number of bolts. 
d t = diameter of bolt. 
D= diameter of bolt circle. 
We will assume that the bolt has the same shearing 
strength as the shaft. The combined shearing strength 
of the bolts is. 7S54:d\nS and their moment of resistance 
to shearing is 

. n^dlnS X |* = . S927 DdlnS 

This last should equal the torsion moment of the 
shaft or .S927 DdtnS=^\ 

Solving for d l and assuming D=2>d as an average 

value, we have d = /^- (79) 

In practice rather larger values are used than would 
be given by the formula. 

70. Shafting Keys. The moment of the shearing 
stress on a key must also equal the twisting moment 
of the shaft. 

Let b= breadth of a key. 
1= length of key. 
h= total depth of key. 
S f = shearing strength of key. 
The moment of shearing stress on key is 
d bdlS' 



blS'X 



2 



Sd* d 

and this must equal -~-y Usually 6=t- 

For shafts of machine steel S=S', and for iron 
shafts $=f £' nearly, as keys should always be of steel. 



140 MACHINE DESIGN. 

Substituting these values and reducing : 
For iron shafting l=1.2d nearly. 

For steel shafting 1=1. Qd nearly, as the least 

lengths of key to prevent its failing 
r-<M by shear. 

O, If the key way is to be designed 
for uniform strength, the shearing 
area of the shaft on the line A B, 
Fig. 63, should equal the shearing 
area of the key, if shaft and key are 
of the same material and AB= 

Fig. 63. CD=h - 

These proportions will make the 
depth of key way in shaft about =|6 and would be 
appropriate for a square key. 

To avoid such a depth of key way which might 
weaken the shaft, it is better to use keys longer than 
required by preceding formulas. In American practice 
the total depth of key rarely exceeds f 6 and one-half 
of this depth is in shaft. 

To prevent crushing of the key the moment of the 
-compressive strength of half the depth of key must 
equal T. 

or 2 X "2 X ^ c= Xl O 

where & is the compressive strength of the key. 

For iron shafts S c = 2S 

3 
and for steel shafts S e =-^S 

Substituting values of S c and assuming h=^b=^d 
we have 

Iron shafts l=2.6d nearly. 

Steel shafts l=Z\d nearly, as the least 

length for flat keys to prevent lateral crushing. 



HANGERS AND BOXES. 



141 



The above refers to parallel keys. Taper keys have 
parallel sides, but taper slightly between top and bot- 
tom. When driven home they have a tendency to tip 
the wheel or coupling on the shaft. This may be par- 
tially obviated by using two keys 90 deg. apart so as 
to give three points of contact between hub and shaft. 
The taper of the keys is usually about { inch to one 
foot. 

The Woodruff key is sometimes used on shafting. 
As may be seen in Fig. 64 this 
key is semi-circular in shape 
and fits a recess sunk in the 
shaft by a milling cutter. 





Fig. 64. 



71. Hangers and Boxes. Since 
shafting is usually hung to the 
ceiling and walls of buildings 
it is necessary to provide means 
for adjusting and aligning the 
bearings as the movement of 
the building disturbs them. 
Furthermore as line shafting 
is continuous and is not per- 
fectly true and straight, the 
bearings should be to a certain 

extent self-adjusting. Eeliable experiments have 
shown that usually one-half of the power developed by 
an engine is lost in the friction of shafting and belts. 
It is important that this loss be prevented as far as 
possible. 

The boxes are in two parts and may be of bored cast- 
iron or lined with Babbitt metal. They are usually 
about four diameters of the shaft in length and are 
oiled by means of a well and rings or wicks. (See Art. 



14:2 



MACHINE DESIGN. 




Fig. 65. 



44.) The best method of supporting the box in the 
hanger is by the ball and socket joint ; all other con- 
trivances such as set screws are but poor substitutes. 

Fig. 65 shows the 
usual arrangement of g| ||^. 

the ball and socket. 

A and B are the two 
parts of the box. The 
center is cast in the 
shape of a partial 
sphere with C as a 
center as shown by 
the dotted lines. The 
two sockets S S can be 
adjusted vertically in 
the hanger by means 

of screws and lock nuts. The horizontal adjustment 
of the hanger is usually effected by moving it bodily 
on the support, the bolt holes being slotted for this 
purpose. 

Counter shafts are short and light and are not subject 

to much bending. Con- 
sequently there is not the 
same need of adjustment 
as in line shafting. 

In Fig. 66 is illustrated 
a simple bearing for 
counters. The solid cast 
iron box B with a spheric- 
al center is fitted directly 
in a socket in the hanger 
H and held in position by 
the cap C and a set screw. 
There is not space here to show all the various forms 




Ij 




l_l 




^0 






// 



B 



Fig. 



HANGERS AND BOXES. 



143 



of hangers and floor stands and reference is made to 
the catalogues of manufacturers. Hangers should be 
symmetrical, i. e., the center of the box should be in a 
vertical line with center of base. They should have 
relatively broad bases and should have the metal dis- 
posed to secure the greatest rigidity possible. Cored 
sections are to be preferred. 




Fig. 67. 



Fie. 68. 



Fig. 67 illustrates the proportions of a Sellers line- 
shaft hanger. This type is also made with the lower 



taking 



down the 



half removable so as to facilitate 
shaft. 

68 shows the outlines of a hanger for heavy 



Fig. 



^^S T^^ssm^ 




Fig. 69. 



shafting as manu- 
factured by the Jones 
& Laughlins Com- 
pany while Fig. 69 il- 
lustrates the design of 
the box with oil wells 
and rings. 



The open side hanger is sometimes adopted on 



EIC- 



144 



MACHINE DESIGN. 



count of the ease with which the shaft can be removed, 

but it is much 
Qi> | { L£l3 less rigid than the 
closed hanger and 
is suitable only 
for light shafting. 
The countershaft 
hanger shown in 
Fig. 70 is simple, 
strong and sym- 
metrical and is a 
great improve- 
ment over those 
Fig. 70. using pointed set 

screws for pivots. 
Hangers similar to this are used by the Brown & 
Sharpe Mfg. Co. with some of their machines. 

Problems. 

1. Calculate the safe diameters of head shaft and three line 
shafts for a factory, the material to be rolled iron and the 
speeds and horse-powers as follows : 




shaft 


100 H P 


200 rev. per min 


ine shop 


30 HP 


120 rev. per min 


rn shop 


50 HP 


250 rev. per min 


3 shop 


20 HP 


200 rev. per min 



2. Determine the horse-power of at least two lines of 
shafting whose speeds and diameters are known. 

3. Design and sketch to scale a flange coupling for a three 
inch line shaft including bolts and keys. 

4. Design a sleeve coupling for the foregoing, different in 
principle from the ones shown in the text. 

5. A four r inch steel head shaft makes 100 rev. per min. 
Find the horse-power which it will safely transmit, and design 
a Weston ring clutch capable of carrying the load. 



HANGERS AND BOXES. 145 

There are to be six wooden rings and five iron rings of 12 in. 
mean diameter. Find the moment carried by each pair of 
surfaces in contact and the end pressure required. ' 

6. Find mean diameter of a single cone clutch for same 
shaft with same end pressure. 

7. Find radial pressure required for a clutch like that 
shown in Fig. 62, the ring being 24 in. in mean diameter and 
there being four pairs of grips. Other conditions as in pre- 
ceding problems. 

8. Select the line shaft hanger which you prefer among 
those in the laboratories and make sketch and description of 
the same. 

9. Do. for a countershaft hanger. 

10. Explain in what way a floor-stand differs from a hanger. 



10 



CHAPTEE X. 

GEARS, PULLEYS AND CRANKS. 

72. Gear Teeth. The teeth of gears may be either 
cast or cut, but the latter method prevails, since cut 
gears are more accurate and run more smoothly and 
quietly. The proportions of the teeth are essentially 
the same for the two classes, save that more back lash 
must be allowed for the cast teeth. The circular pitch 
is obtained by dividing the circumference of the pitch 
circle by the number of teeth. The diametral pitch is 
obtained by dividing the number of teeth by the diam- 
eter of the pitch circle and equals the number of teeth 
per inch of diameter. The reciprocal of the diametral 
pitch is sometimes called the module. The addendum 
is the radial projection of the tooth beyond the pitch 
circle, the dedendum the corresponding distance inside 
the pitch circle. The clearance is the difference be- 
tween the dedendum and addendum; the back lash 
the difference between the widths of space and tooth 
on the pitch circle. 

Let circular pitch = p. 

module = ™=m. 

7T 

diametral pitch =— = — 
r p m 

addendum =a. 

dedendum or flank =/. 

clearance =f—a=c. 

height =a+f=h. 

width = w. (See Fig. 72.) 

146 



GEAR TEETH. 147 

The usual rule for standard cut teeth is to make 
w = £ allowing no calculable back-lash, to make a=m 

and f——Fr or h=2h7i and clearance = 7r - 

J 8 8 

There is however a marked tendency at the present 
time towards the use of shorter teeth. The reasons 
urged for their adoption are : first, greater strength 
and less obliquity of action ; second, less expense in 
cutting.* Several systems have been proposed in which 
the height of tooth h varies from 0.425p to 0.55p. 

According to the latter system a=0.25p, /=0.3p, 
and c=.05p. 

In modern practice the diametral pitch is a whole 
number or a common fraction and is used in describing 
the gear. For instance a 3 pitch gear is one having 
3 teeth per inch of diameter. The following table 
gives the pitches in common use and the proportions 
of long and short teeth. 

If the gears are cut, w=£ ; if cast gears are used, 
w=0A6p to OASp. 

* See American Machinist, Jan. 7, 1897, p. 6. 



148 



MACHINE DESIGN. 



TABLE XXVI. 

Proportions of Gear Teeth. 



Pitch. 


Standard Teeth. 


Short Teeth. 


Diame- 
tral. 


Circular. 


Addend. 
a 


Height. 
h 


Clear- 
ance, 
c 


Addend. 
a 


Height. 
h 


Clear- 
ance, 
c 


\ 


6.283 


2. 


4.25 


0.25 


1.571 


3.456 


0.314 


I 


4.189 


1.33 


2.82 


0.167 


1.047 


2.303 


0.209 


1 


3.142 


1. 


2.125 


0.125 


0.785 


1.728 


0.157 


H 


2.513 


0.8 


1.7 


0.1 


0.628 


1.383 


0.125 


n 


2.094 


0.667 


1.415 


0.083 


0.524 


1.152 


0.105 


if 


1.795 


0.571 


1.212 


0.071 


0.449 


0.988 


0.09 


2 


1.571 


0.5 


1-062 


0.062 


0.392 


0.863 


0.078 


2± 


1.396 


0.445 


0.945 


0.056 


0.349 


0.768 


0.070 


2£ 


1.257 


0.4 


0.85 


0.05 


0.314 


0.691 


0.063 


2f 


1.142 


0.364 


0.775 


0.045 


0.286 


0.629 


0.057 


3 


1.047 


0.333 


0.708 


0.042 


0.262 


0.576 


0.052 


3i 


0.898 


0.286 


0.608 


0.036 


0.224 


0.494 


0.045 


4 


0.785 


0.25 


0.531 


0.031 


0.196 


0.432 


0.039 


5 


0.628 


0.2 


0.425 


0.025 


0.157 


0.345 


0.031 


6 


0.524 


0.167 


0.354 


0.021 


0.131 


0.288 


0.026 


7 


0.449 


0.143 


0.304 


0.018 


0.112 


0.246 


0.022 


8 


0.393 


0.125 


0.266 


0.016 


0.098 


0.216 


0.020 


9 


0.349 


0.111 


0.236 


0.014 


0.087 


0.191 


0.017 


10 


0.314 


0.1 


0.212 


0.012 


0.079 


0.174 


0.016 


11 


0.286 


0.091 


0.193 


0.011 


0.071 


0.156 


0.014 


12 


0.262 


0.0834 


0.177 


0.010 


0.065 


0.143 


0.013 


13 


0.242 


0.077 


0.164 


0.010 


0.060 


0.132 


0.012 


14 


0.224 


0.0715 


0.152 


0.009 


0.056 


0.123 


0.011 


15 


0.209 


0.0667 


0.142 


0.008 


0.052 


0.114 


0.010 


16 


0.196 


0.0625 


0.133 


0.008 


0.049 


0.108 


0.010 



73. Strength of Teeth. 

Let P = total driving pressure on wheel at pitch cir- 
cle. This may be distributed over two or more teeth, 
but the chances are against an even distribution. 

Again, in designing a set of gears the contact is 
likely to be confined to one pair of teeth in the smaller 
pinions. 



STRENGTH OF TEETH. 



149 



Each tooth should therefore be made strong enough 
to sustain the whole pressure. 

Rough Teeth. The teeth of pattern molded gears are 
apt to be more or less irregular in shape, and are 
especially liable to be thicker at one end on account of 
the draft of the pattern. 

In this case the entire pressure may come on the 
outer corner of a tooth and tend to cause a diagonal 
fracture. 

Let C in Fig. 71 be the point of application of the 
pressure P, and AB the line of probable fracture. 



Drop the 

1 CD on 
AB 

Let AB = x 

and 

CD = y 

angle 

CAD = a 






Fig. 72. 

strength of material. 

Py=. 



c 



A 



Fig. 71. 

The bending moment at 
section AB is M=Py, and 
the moment of resistance is 

M =-Lsxw 2 
b 

were S = safe transverse 



Sxw' 



and 



S= 



GFy. 



(a) 



150 MACHINE DESIGN. 

If P and w are constant, then S is a maximum 

when -^- is a maximum. 
a? 

But y = h sina and x= 

cosa 



1 

X 

maximum when a = 45° and $- = 



— =» cosa which is a 

X 



X 



op 

Substituting this value in (a) we ha,veS=— 

3 P 
But in this case w =A7p and therefore £-— ' 



.221p 2 

and p=3.684\S" (80) 

diametral pitch, — =.853 /^ (81) 

Unless machine molded teeth are very carefully 
made, it may be necessary to apply this rule to them 
as well. 

Cut Gears. With careful workmanship machine 
molded and machine cut teeth should touch along the 
whole breadth. In such cases we may assume a line 
of contact at crest of tooth and a maximum bending 
moment. 

M=Ph 

The moment of resistance at base of tooth is 

M 1 = \Sbw' 

when b is the breadth of tooth. 

In most teeth the thickness at base is greater than 
w, but in radial teeth it is less. Assuming standard 
proportions for cut gears : 



LEWIS' FORMULAS. 151 



h = 2|w=.6765p 
w = .5p 
and substituting above : 



.6765 Pp^ 

P=M16bSp ....... (82) 

For short teeth having h = .55p formula (82) reduces 
to: 

P = .075$bSp (83) 

The above formulas are general whatever the ratio 
of breadth to pitch. The general practice in this coun- 
try is to make 

b = Sp 

Substituting this value of b in (82) and (83) and 
reducing : 
Long teeth : p = 2.326 J*L •••-•.•• (84) 

Short teeth :p = 2.098 J| /. . .(85) 

The corresponding formulas for the diametral pitch 
are : 
Long teeth : L = 1.35 ^ ■ • • ( 86 ) 

Short teeth:— =1.49 J& ( 8 ^) 

m \p 

74. Lewis' Formulas. The foregoing formulas can 
only be regarded as approximate, since the strength 
of gear teeth depends upon the number of teeth in the 
wheel ; the teeth of a rack are broader at the base and 
consequently stronger than those of a pinion. This 
is more particularly true of epicycloidal teeth. Mr. 
Wilfred Lewis has deduced formulas which take into 



152 MACHINE DESIGN. 

account this variation. For cut spur gears of standard 
dimensions the Lewis formula is as follows : 



P = &£p(.124-^) (88) 

where n= number of teeth. 

This formula reduces to the same as (82), for n=14 
nearly. 

Formula (82) would then properly apply only to 
small pinions, but as it would err on the safe side for 
larger wheels, it can be used where great accuracy is 
not needed. The same criticism applies to the other 
formulas in Art. 73. 

The value of S used should depend on the material 
and on the speed. 

The following safe values are recommended for cast 
iron and cast steel. 



Linear velocity 
ft. per min. 


100 


200 


300 


600 


900 


1200 


1800 


2400 


Cast Iron 

Cast Steel 


8000 
24000 


6000 
15000 


4800 
12000 


4000 
10000 


3000 
7500 


2400 
6000 


2000 
5000 


1700 
4250 



For gears used in hoisting machinery where there is 
slow speed and liability of shocks a writer in the 
American Machinist recommends smaller values of S 
than those given above * and proposes the following 
for four different metals : 



* American Machinist, Feb. 16, 1905. 



EXPERIMENTAL DATA. 153 



Linear velocity 
ft. per miD. 


100 


200 


300 


600 


900 


1200 


1800 


2400 


Gray Iron 


4800 


4200 


3840 


3200 


2400 


1920 


1600 


1360 


Gun Metal 


7200 


6300 


5760 


4800 


3600 


2880 


2400 


2040 


Cast Steel 


9600 


8400 


7680 


6400 


4800 


3840 


3200 


2720 


Mild Steel 


12000 


10500 


9600 


8000 


6000 


4800 


4000 


3400 



The experiments described in the next article show- 
that the ultimate values of S are much less than the 
transverse strength of the material and point to the 
need of large factors of safety. 

75. Experimental Data. In the American Machinist 
for Jan. 14, 1897, are given the actual breaking loads 
of gear teeth which failed in service. The teeth had 
an average pitch of about 5 inches, a breadth of about 
18 inches and the rather unusual velocity of over 
2000 ft. per minute. The average breaking load was 
about 15000 lb. there being an average of about 50 
teeth on the pinions. Substituting these values in 
(88) and solving we get 

£=1575 lb. 

This very low value is to be attributed to the con- 
dition of pressure on one corner noted in Art. 73. 
Substituting in formula for such a case. 

.221p 2 

This all goes to show that it is well to allow large 
factors of safety for rough gears, especially when the 
speed is high. 

Experiments have been made on the static strength 



154 MACHINE DESIGN. 

of rough cast iron gear teeth at the Case School of 
Applied Science by breaking them in a testing machine. 
The teeth were cast singly from patterns, were two 
pitch and about 6 inches broad. The patterns were 
constructed accurately from templates representing 
15 deg. involute teeth and cycloidal teeth drawn with a 
describing circle one-half the pitch circle of 15 teeth ; 
the proportions used were those given for standard cut 
gears. 

There were in all 41 cycloidal teeth of shapes cor- 
responding to wheels of 15-24-36-48-72-120 teeth and 
a rack. There were 28 involute teeth corresponding to 
numbers above given omitting the pinion of 15 teeth. 

The pressure was applied by a steel plunger tangent 
to the surface of tooth and so pivoted as to bear evenly 
across the whole breadth. The teeth were inclined at 
various angles so as to vary the obliquity from to 25 
deg. for the cycloidal and from 15 deg. to 25 deg. for 
the involute. The point of application changed accord- 
ingly from the pitch line to the crest of the tooth. 
From these experiments the following conclusions are 
drawn : 

1. The plane of fracture is approximately parallel to 
line of pressure and not necessarily at right angles to 
radial line through center of tooth. 

2. Corner breaks are likely to occur even when the 
pressure is apparently uniform along the tooth. There 
were fourteen such breaks in all. 

3. With teeth of dimensions given, the breaking 
pressure per tooth varies from 25000 lb. to 50000 lb. 
for cycloids as the number of teeth increases from 15 
to infinity ; the breaking pressure for involutes of the 
same pitch varies from 34000 lb. to 80000 lb. as the 
number increases from 24 to infinity. 



TEETH OF BEVEL GEARS. 155 

4. With teeth as above the ayerage breaking pres- 
sure varies from 50000 lb. to 26000 lb. in the cycloids 
as the angle changes from deg. to 25 deg. and the 
tangent point moves from pitch line to crest ; with 
involute teeth the range is between 64000 and 39000 lb. 

5. Eeasoning from the figures just given, rack teeth 
are about twice as strong as pinion teeth and involute 
teeth have an advantage in streDgth over cycloidal of 
from forty to fifty per cent. The advantage of short 
teeth in point of strength can also be seen. The 
modulus of rupture of the material used was about 
36000 lb. Values of S calculated from Lewis' formula 
for the various tooth numbers are quite uniform and 
average about 40000 lb. for cycloidal teeth. Involute 
teeth are to-day generally preferred by manufacturers. 
William Sellers & Co. use an obliquity of 20 deg. in- 
stead of 14J or 15 deg. the usual angle. 

76. Teeth of Bevel Gears. There have been many 
formulas and diagrams proposed for determining the 
strength of bevel gear teeth, some of them being very 
complicated and inconvenient. It will usually answer 
every purpose from a practical standpoint, if we treat 
the section at the middle of the breadth of such a tooth 
as a spur wheel tooth and design it by the foregoing 
formulas. The breadth of the teeth of a bevel gear 
should be about one-third of the distance from the base 
of the cone to the apex. 

One point needs to be noted ; the teeth of bevel 
gears are stronger than those of spur gears of the 
same pitch and number of teeth since they are developed 
from a pitch circle having an element of the normal 
cone as a radius. To illustrate, we will suppose that 
we are designing the teeth of a miter gear and that 



156 MACHINE DESIGN. 

the number of teeth is 32. In such a gear the element 
of normal cone is |/ 2 times the radius. The actual 
shape of the teeth will then correspond to those of a 
spur gear having 32 j/ 2=45 teeth nearly. 

Note. — In designing the teeth of gears where the 
number is unknown, the approximate dimensions may 
first be obtained by formula (84) or (85) and then these 
values corrected by using Lewis' formula. 

Problems. 

1. The drum of a hoist is 8 in. in diameter and makes 5 
rev. per minute. The diameter of gear on the drum is 36 
inches and of its pinion 6 in. The gear on the counter shaft 
is 24 in. in diameter and its pinion is 6 in. in diameter. The 
gears are all cut. 

Calculate the pitch and number of teeth of each gear, as- 
suming a load of one ton on drum chain and 5=6000. Also 
determine the horse-power of the machine. 

2. Calculate the pitch and number of teeth of a cut cast 
steel gear 10 in. in diameter, running at 250 rev. per min. 
and transmitting 20 HP. 

3. A cast-iron gear wheel is 30 ft. 6§ in. in pitch diameter 
and has 192 teeth, which are machine-cut and 30 in. broad. 

Determine the circular and diameter pitches of the teeth and 
the horse-power which the gear will transmit safely when 
making 12 rev. per min. 

4. A two pitch cycloidal tooth, 6 in. broad, 72 teeth to the 
wheel, failed under a load of 38000 lb. Find value of S by 
Lewis' formula. 

5. A vertical water-wheel shaft is connected to horizontal 
head shaft by cast iron gears and transmits 150 HP. The 
water-wheel makes 200 rev. per min. and the head shaft 100. 

Determine the dimensions of the gears and teeth if the latter 
are approximately two pitch. 

6. Work Problem 1, using short teeth instead of standard. 



RIM AND ARMS. 157 

77. Rim and Arms. The rim of a gear, especially 
if the teeth are cast, should have nearly the same 
thickness as the base of tooth, to avoid cooling strains. 

It is difficult to calculate exactly the stresses on 
the arms of the gear, since we know so little of the 
initial stress present, due to cooling and contraction. 
A hub of unusual weight is liable to contract in cooling 
after the arms have become rigid and cause severe 
tension or even fracture at the junction of arm and 
hub. 

A heavy rim on the contrary may compress the arms 
so as actually to spring them out of shape. Of course 
both of these errors should be avoided, and the pattern 
be so designed that cooling shall be simultaneous in 
all parts of the casting. 

The arms of spur gears are usually made straight 
without curves or taper, and of a flat, elliptical cross- 
section, which offers little resistance to the air. To 
support the wide rims of bevel gears and to facilitate 
drawing the pattern from the sand, the arms are some- 
times of a rectangular or T section, having the greatest 
depth in the direction of the axis of the gear. For 
pulleys which are to run at a high speed it is important 
that there should be no ribs or projections on arms or 
rim which will offer resistance to the air. Experiments 
by the writer have shown this resistance to be serious 
at speeds frequently used in practice. 

A series of experiments conducted by the author are 
reported in the American Machinist for Sept. 22, 1898, 
to which paper reference is here made. 

Twenty-four pulleys having 3^- inches face and 
diameters of 16, 20 and 2<± inches were broken in a 
testing machine by the pull of a steel belt, the ratio of 
the belt tensions being adjusted by levers so as to be 



158 MACHINE DESIGN. 

two to one. Twelve of the pulleys were of the ordi- 
nary cast-iron type having each six arms tapering and 
of an elliptic section. The other twelve were Meclart 
pulleys with steel rims riveted to arms and having 
some six and some eight arms. Test pieces cast from 
the same iron as the pulleys showed an average modu- 
lus of rupture of 35800 for the cast-iron and 50800 for 
the Medart. 

In every case the arm or the two arms nearest the 
side of the belt having the greatest tension, broke first, 
showing that the torque was not evenly distributed by 
the rim. Measurements of the deflection of the arms 
showed it to be from two to six times as great on this 
side as on the other. The buckling and springing of 
the rim was very noticeable especially in the Medart 
pulleys. 

The arms of all the pulleys broke at the hub showing 
the greatest bending moment there, as the strength of 
the arms at the hub was about double that at the rim. 
On the other hand some of the cast iron arms broke 
simultaneously at hub and rim, showing a negative 
bending moment at the rim about one-half that at the 
hub. 

The following general conclusions are justified by 
these experiments : 

(a) The bending moments on pulley arms are not 
evenly distributed by the rim, but are greatest next 
the tight side of belt. 

(b) There are bending moments at both ends of 
arm, that at the hub being much the greater, the ratio 
depending on the relative stiffness of rim and arms. 

The following rules may be adopted for designing 
the arms of cast iron pulleys and gears : 

1. Multiply the net turning pressure, whether caused 



SPROCKET WHEELS AND CHAINS. 159 

by belt or tooth, by a suitable factor of safety and by 
the length of the arm in inches. Divide this product 
by one-half the number of arms and use the quotient 
for a bending moment. Design the hub end of arm to 
resist this moment. 

2. Make the rim ends of arms one-half as strong as 
the hub ends. 

78. Sprocket Wheels and Chains. Steel chains con- 
necting toothed wheels afford a convenient means of 
getting a positive speed ratio when the axes are some 
distance apart. There are three classes in common 
use, the block chain, the roller chain and the so-called 
" silent " chain. 

Mr. A. Eugene Michel publishes quite a complete 
discussion of the design of the first two classes in 
Machinery for February, 1905, and reference is here 
made to that journal. 

Block chain is that commonly used on bicycles and 
small motor cars, so named from the blocks with 
round ends which are 
used to fill in between 
the links. The sprocket 
teeth are spaced to a 
pitch greater than that 
of the chain links and 
the blocks rest on flat 
beds between the teeth, 
Fig. Y3. 

Roller chains have 
rollers on every pin and have inside and outside links. 
The sprocket teeth have the same pitch as the chain 
links, the rollers fitting circular recesses between the 
sprockets, Fig. 74. 




160 



MACHINE DESIGN. 



The most serious failing of the chain is its tendency 

to stretch with use so that the pitch becomes greater 

than that of the sprocket teeth. 

To obviate this difficulty in a measure considerable 

clearance should be 
given to the sprocket 
teeth as indicated in 
Fig. 74. As the pitch 
of the chain increases 
it will then ride higher 
upon the sprockets 
until the end of the 
tooth is reached. The 
teeth are rounded on 

their side faces, that they may easily enter the gaps in 

the chain and have side clearance. 

Mr. Michel gives the following values for the tensile 

strength of chains as determined by actual tests. 

Roller Chain. 




Fig. 74. 



Pitch inches 


i 

2 


5 
8" 


3 

I" 


1 


1* 


H 


1 3 


2 


Tensile 


















Strength lb. 


1200 


1200 


4000 


6000 


9000 


12000 


19000 


25000 



Block Chain. 

1 inch pitch 1200 to 2500 lb. 
1% " ' " 5000 " 

Mr. Michel further recommends a factor of safety 
of from 5 to 40 according to the severity of the condi- 
tions as to speed and shocks. 

The tendency is to use short links and double or 
triple width chains to increase the rivet bearing sur- 



SILENT CHAINS. 161 

face, as it is this latter factor which really determines 
the life of a chain. 

Eoller chains may be used up to speeds of 1000 to 
1200 feet per minute. 

The sprocket should be so designed that one tooth 
will carry the load safely with the pressure near the 
crest since these conditions obtain as the chain stretches. 
Use values of S as in Art. 74. 

79. Silent Chains. The weak points in the ordinary 
chain, whether it be made with blocks or rollers, are 
the rivet bearings. It is the continual wear of these, 
due to insufficient area and lack of proper lubrication, 
that shortens the life of a chain. 

The so-called "silent- 
chain " with rocker bear- 
ings, is comparatively 
free from this defect. 
Fig. 75 illustrates the 
shapes of links, rivets 
and sprockets for this 
kind of chain as man- 
ufactured by the Morse Chain Company. 

The chain proper is entirely outside of the sprocket 
teeth so that the latter may be continuous across the 
face of the wheel, save for a single guiding groove in 
the center. 

Projections on the under side of the links engage 
with the teeth of the sprocket, E being the point of 
contact for the driver and / a similar point for the 
follower when the rotation is as indicated. 

Each rivet consists practically of two pins called by 

the makers the rocker pin and the seat pin. Each pin 

is fastened in its particular gang of links and the 
11 




162 



MACHINE DESIGN. 



relative motion is merely a rocking of one pin on the 
other without appreciable friction. 

The pins are of hardened tool steel with softened 
ends. The combination of this freedom from rubbing 
contact with the adaptation of the engaging tooth 
profiles, gives a chain which can be safely run at high 
speeds without objectionable vibration or appreciable 
wear. 

The chains can be made of almost any width from 
one-half inch up to eighteen inches, the width de- 
pending upon the pitch of the chain and the power to 
be transmitted. 

The following are the working loads (and limiting 
speeds) of chains two inches in width and of different 
pitches, taken from a table published by the makers : 



Pitch in inches 


i 

2 


5 
8 


3 


.9 


1.2 


1.5 


Working load 
in pounds 


130 


190 


236 


380 


520 


760 


Limiting Speed 
Eev. per min. 


2000 


1600 


1200 


1100 


800 


600 



The number of teeth in the small sprocket may vary 
from 15 to 30 according to the conditions. 

Assuming 17 teeth and the number of revolutions 
given in the above table the speed of chain would be 
1420 feet per minute for the \ inch pitch and 1275 feet 
per minute for the 1.5 inch. 

Chains of this character have been run successfully 
at 2000 feet per minute. 



CRANKS AND LEVERS. 



163 



Problems. 



1. Design eight arms of elliptic section for a gear 48 inches 
pitch diameter, to transmit a pressure on tooth of 800 pounds. 
Material, cast iron having a working transverse strength of 
6000 pounds per square inch. 

2. Two sprocket wheels of 75 and 17 teeth respectively are 
to transmit twenty horse-power at a chain speed of about 800 
feet per minute, with a factor of safety of 12 — 

Determine the proper pitch of roller chain, the pitch diam- 
eters of the sprockets, and the numbers of revolutions. 

3. Suppose that in Problem 2, a "silent" chain is to be 
used and the chain speed increased to 1200 feet per minute. 
Determine the proper pitch of chain to be used if the width of 
chain is 3 inches. Determine diameters and revolutions of 
sprockets as before. 

Cranks and Levers. A crank or rocker arm which 
is used to transmit a continuous or reciprocating 
rotary motion is in the condition of a cantilever or 
bracket with a load at the outer end. 

If the web of the crank is of uniform thickness 
theory requires that its profile should be parabolic for 
uniform strength, the vertex of the parabola being at 
the load point. 

A convenient approximation to this shape can be 



attained by 



the tangents to the parabola at 




Fig. 76. 
points midway between the hub and the load point. 
See Fig. 76, The crank web is designed of the right 



164 MACHINE DESIGN. 

thickness and breadth to resist the moment at AB, 
and the center line is produced to Q, making PQ = 4 
PO. 

Straight lines drawn from Q to A and B will be 
tangent to the parabola at the latter points and will 
serve as contour lines for the web. 

Assume the following dimensions in inches : 

I = length of crank = OP. 

t = thickness of web. 

h = breadth " " = AB. 

d = diameter of eye = cd. 

d 1= " " pin. 

b = breadth of eye. 

D = diameter of hub = CD. 

D,= " " shaft. 

B = breadth of hub. 

If the pressure on the crank pin is denoted by P 

PI 

then will the moment at AB be — - and the equa- 

tions of moments for the cross-section will be : 

Pl^ Sth* 

2 6 [See Formula (3)] 

and from this the dimensions at AB maybe calculated. 
The moment at the hub will be PI and will tend to 
break the iron on the dotted lines CD. The equation 
of moments for the hub is therefore : 

From this equation the dimensions of the hub may 
be calculated when D ± is known. The eye of a crank is 
most likely to break when the pressure on the pin is 
along the line OP, and the fracture will be along the 
dotted lines cd. The bending moment will be Pmul- 



CRANKS AND LEVERS, 165 

tiplied by the distance from center of pin to center of 
eye measured along axis of pin. If we call this dis- 
tance x, then will the equation of moments be : 

o 
It is considered good practice among engine builders 
to make the values of x, b and B as small as practicable, 
in order to reduce the twisting moment on the web 
of the crank and the bending moment on the shaft. In 
designing the hub, allowance must be made for the 
metal removed at the key- way. 

Problem. 

Design a cast steel crank for a steam engine having a cylin- 
der 12 by 30 inches and an initial steam pressure of 120 lb. 
per sq. in. of piston. The shaft is 6 inches and the crank pin 
3 inches in diameter. The distance x may be assumed as 4 
inches. Calculate, 

1. Dimensions of web at AB. 

2. Dimensions of hub allowing for a key 1 x f- inches. 

3. Dimensions of eye for pin, make a scale drawing in ink 
showing profile of crank complete, £may be assumed as 6,000 
lb. per sq. in. 



CHAPTER XI. 

FLY-WHEELS. 

81. In General. The hub and arms of a fly-wheel 
are designed in much the same way as those of pulleys 
and gears, the straight arm with elliptic section being 
the favorite. The rims of such wheels are of two 
classes, the wide, thin rim used for belt transmission 
and the narrow solid rim of the generator or blowing 
engine wheel. Fly-wheels up to eight or ten feet in 
diameter are usually cast in one piece ; those from ten 
to sixteen feet in diameter may be cast in halves, while 
wheels larger than the last mentioned should be cast 
in sections, one arm to each section. 

This is a matter, not of use, but of convenience in 
transportation. 

The joints between hub and arms and between arms 
and rim need not be specially considered here, since 
wheels rarely fail at these points. 

The rim and the joints 
in the rim cannot be too 
carefully designed. The 
smaller wheel cast in one 
piece is more or less sub- 
ject to stresses caused by 
shrinkage. The sectional 
Fig. 77. wheel is generally free 

from such stresses but is 
weakened by the numerous joints. 

Rim joints are of two general classes according as 
bolts or links are used for fastenings, 

166 




SAFE SPEED FOR WHEELS. 



167 



"Wide, thin rims are usually fastened together by 
internal flanges and bolts as shown in Fig. 77, while 
the stocky rims of the fly-wheels proper are joined 
directly by links or T— head "prisoners " as in Fig. 78. 

As will be shown 
later, the former is a 
weak and unreliable 
joint, especially 
when located mid- 
way between the 
arms. 

The principal 
stresses in fly-wheel 
rims are caused by centrifugal force 




Fig. 78. 



82. Safe Speed for Wheels. The centrifugal force 
developed in a rapidly revolving pulley or gear pro- 
duces a certain tension on the rim, and also a bending 
of the rim between the arms. We will first investigate 
the case of a pulley having a rim of uniform cross 
section. 

It is safe to assume that the rim should be capable 
of bearing its own centrifugal tension without assist- 
ance from the arms. 

Let D=mean diameter of pulley rim. 
t= thickness of rim. 
b= breadth of rim. 
iv— weight of material per cu. in. 

= .26 lb. for cast-iron. 

= .28 lb. for wrought iron or steel. 
n— number of arms. 
N= number rev. per min. 
r= velocity of rim in ft. per sec. 



108 MACHINE DESIGN. 

First let us consider the centrifugal tension alone. 
The centrifugal pressure per square inch of concave 
surface is 

p= .... (a) 

where W is the weight of rim per square inch of con- 
cave surface =wt, and r= radius in feet =^-r* 

24 

The centrifugal tension produced in the rim by this 
force is by formula (13) 

2t 
Substituting the values of p, W and r and reducing : 

S=^™1 .... (89) 

and v-jj* .... (90) 

For an average value of w=.27 9 (89) reduces to 

S=jq nearly. 

a convenient form to remember. 

The corresponding values of S for dry wood and for 
leather would be nearly : 

Wood 8^. 

Leather $=37V 

If we assume S as the ultimate tensile strength, 
16500 lbs. for cast-iron in large castings and 60000 lbs. 
for soft steel, then the bursting speed of rim is : 
for a cast-iron wheel ^=406 ft. per sec. . (91) 
and for steel rim v=775 ft. per sec. * . (92) 

and these values may be used in roughly calculating 
the safe speed of pulleys. 



SAFE SPEED FOR WHEELS. 169 

It has been shown by Mr. James B. Stanwood, in a 
paper read before the American Society of % Mechanical 
Engineers,* that each section of the rim between the 
arms is moreover in the condition of a beam fixed at 
the ends and uniformly loaded. 

This condition will produce an additional tension 
on the outside of rim. The formula for such a beam 
when of rectangular cross-section is 

12 "" 6 * ' ' * w 

W in this case is the centrifugal force of the fraction 
of rim included between two arms. 

The weight of this fraction is — and its cen- 

trifugal force W= ^btw 2^ or wJ*&Z* 

n gD gn 

Also Z=- — and d=t 

n 

Substituting these values in (b) and solving for S : 

S=3S18*^ (c) 

tn 2 K J 

If iv is given an average value of .27 then 

S =-Jtf~ nearly (d) 

and the total value of the tensile stress on outer sur- 
face of rim is 

S'=?g+ ^-nearly (93) 



Solving for v 

r ( 9i ) 

fri 2_r 10 
In a pulley with a thin rim and small number of 






* See Trans. A. S. M. E. Vol. XIV. 



170 MACHINE DESIGN. 

arms, the stress due to this bending is seen to be con- 
siderable. 

It must, however, be remembered that the stretching 
of the arms due to their own centrifugal force and that 
of the rim will diminish this bending. Mr. Stanwood 
recommends a deduction of one-half from the value of 
S in (d) on this account. 

Prof. Gaetano Lanza has published quite an elab- 
orate mathematical discussion of this subject. (See 
Vol. XVI. Trans. Am. Soc. Mech. Engineers.) He 
shows that in ordinary cases the stretch of the arms 
will relieve more than one-half of the stress due to 
bending, perhaps three-quarters. 

83. Experiments on Fly- Wheels. In order to de- 
termine experimentally the centrifugal tension and 
bending in rapidly revolving rims, a large number of 
small fly-wheels have been tested to destruction at the 
Case School laboratories. In all ten wheels, fifteen 
inches in diameter and twenty-three wheels two feet in 
diameter have been so tested. An account of some of 
these experiments may be found in Trans. Am. Soc. 
Mech. Eng. Vol. XX. The wheels were all of cast- 
iron and modeled after actual fly-wheels. Some had 
solid rims, some jointed rims and some steel spokes. 

To give to the wheels the speed necessary for de- 
struction, use was made of a Dow steam turbine capa- 
ble of being run at any speed up to 10000 revolutions 
per minute. The turbine shaft was connected to the 
shaft carrying the fly-wheels by a brass sleeve coup- 
ling loosely pinned to the shafts at each end in such a 
way as to form a universal joint, and so proportioned 
as to break or slip without injuring the turbine in case 
of sudden stoppage of the fly-wheel shaft. 



EXPERIMENTS ON FLY-WHEELS. 



171 



One experiment with a shield made of two-inch 
plank proved that safety did not lie in that direction, 
and in succeeding experiments with the fifteen inch 
wheels a bomb-proof constructed of 6X12 inch white 
oak was used. The first experiment with a twenty- 
four inch wheel showed even this to be a flimsy contri- 
vance. In subsequent experiments a shield made of 
12x12 inch oak was used. This shield was split re- 
peatedly and had to be re- enforced by bolts. 

A cast steel ring about four inches thick lined, with 
wooden blocks and covered with three inch oak plank- 
ing, was finally adopted. 

The wheels were usually demolished by the ex- 
plosion. No crashing or rending noise was heard, 
only one quick, sharp report, like a musket shot. 

The following tables give a summary of a number 
of the experiments. 

TABLE XXVI. 
FIFTEEN INCH WHEELS. 





Bursting Speed. 


Centrifugal 
Tension 

~10 




No. 






Remarks. 




Rev. 


Feet per 






per Minute. 


Second =v. 




1 


6,525 


430 


18,500 


Six arms. 


2 


6,525 


430 


18,500 


Six arms. 


3 


6,035 


395 


15,600 


Thin rim. 


4 


5,872 


380 


14,400 


Thin rim. 


5 


2,925 


192 


3,700 


Joint in rim. 


6 


5,600* 


368 


13,600 


Three arms. 


7 


6,198 


406 


16,500 


Three arms. 


8 


5,709 


368 


13,600 


Three arms. 


9 


5,709 


365 


13,300 


Thin rim. 


10 


5,709 


361 


13,000 


Thin rim. 



* Doubtful. 



172 



MACHINE DESIGN. 



TABLE XXVII. 

TWENTY-FOUR INCH WHEELS. 





Shape and Size of Rim. 


Weight 

of 
Wheel, 


6 


Diam- 


Breadth 


Depth 


Area 






eter 
Inches. 


Inches. 


Inches. 


Sq. 
Inches. 


Style of Joint. 


Pounds 


11 


24 


Si 


1.5 


3.18 


Solid rim. 


75.25 


12 


24 


±h 


.75 


3.85 


Internal flanges, bolted 


93. 


13 


24 


4 


.75 


3.85 


<( (< ti 


91.75 


14 


24 


4 


.75 


3.85 


a << tt 


95. 


15 


24 


4yV 


.75 


3.85 


<< (< «< 


94.75 


16 


24 


o 


2.1 


2.45 


Three lugs and links. 


65.1 


17 


24 


1.2 


2.1 


2.45 


Two lugs and links. 


65. 



TABLE XXVIII. 

Flanges and Bolts. 







Flanges. 






Bolts. 






Thickness. 


Effective 


Effective 


No. to each 


Diameter. 


Total 


No 




Breadth. 


Area. 






Tensile 














Strength. 




Inches. 


Inches. 


Inches. 






Pounds. 


12 


n 


2.8 


1.92 


4 


T 5 6 


16,000 


13 


tt 


2.75 


1.89 


4 


x 5 <r 


16,000 


14 


it 


2.75 


2.58 


4 


i 5 * 


16,000 


15 


15 


2.5 


2.34 


4 


t 


20,000 



BY TESTING MACHINE. 



Tensile strength of cast-iron =19,600 pounds per square in. 
Transverse strength of cast-iron =46, 600 pounds per square in. 
Tensile strength of •& bolts =4,000 pounds. 
Tensile strength of f bolts =5,000 pounds. 



EXPERIMENTS ON FLY-WHEELS. 



173 



TABLE XXVIX. 

Failure of Flanged Joints. 







to 




Bursting 


Cent. 






CO 


e3 

03 C 

s- 02 


"So *§ 

2 o 

02 ^ 


Speed. 


Tension. 




No. 


Rev. 


Ft. per 


Per 




Remarks. 




«8 53 

CD CO 


03 03 


c3 t» 


per 


Sec. 


Sq. In. 


Total 






^ 02 

3.18 


m a 

m c3 




Min. 


=v 


10 


Lbs. 




11 






3,672 


385 


14,800 


47,000 


Solid rim. 


12 


3.85 


1.92 


16,000 




• . • • 


.... 




Flange broke. 


13 


3.85 


1.89 


16,000 


1,760 


184 


3,400 


13,100 


Flange broke. 


14 


3.85 


2.58 


16,000 


1,875 


196 


3,850 


14,800 


Bolts broke. 


15 


3.85 


2.34 


20,000 


1,810 


190 


3,610 


13,900 


Flange broke. 



TABLE XXX. 

Linked Joints. 





Lugs. 


Links. 


Rim. 


No. 


DO 


w 

03 


d 


•d 


^ 


8 


c3 


Max. 


Net 




.£3 A 

-a a 


US 




S 


T3 co 
^> CO 03 
O 03 J3 
03 £ 13 


a » 


|3- 

"8 5 


Area, 


Area, 






03 


2 

< 


3 


43 M a 


£° 


H 02 


Sq. Ins. 


Sq. Ins. 


16 


.45 


1.0 


.45 


3 


.57 


.327 


.186 


2.45 


1.98 


17 


.44 


.98 


.43 


2 


.54 


.380 


.205 


2.45 


1.98 



BY TESTING MACHINE. 

Tensile strength of cast-iron =19,600. 
Transverse strength of cast-iron =40, 400. 
Av. tensile strength of each link= 10,180. 



174 



MACHINE DESIGN. 



TABLE XXXI. 

Failure of Linked Joints. 





w 
M 

a 

" CO 

o a 
•* 2 

-u o 
bp pn 
a 

2 
5 


i 

o a 
S o 

W) P-l 

a 

0) 
h 

+3 

BO 


Bursting 
Speed. 


Cent. Tension. 




No. 


Rev. 
per 
Min. 


Ft. per 
Sec. 
=v 


Per 
Sq_. In. 

V 2 

= 10 


Total. 


Remarks. 


16 
17 


30,540 
20,360 


38,800 
38,800 


3,060 
2,750 


320 

290 


10,240 
8,410 


25,100 
20,600 


Rim broke. 
Lugs and Rim 
broke. 



The flanged joints mentioned had the internal flanges 
and bolts common in large belt wheel rims while the 
linked joints were such as are common in fly-wheels 
not used for belts. 

* Subsequent experiments have given approximately 
the same results as those just detailed. The highest 
velocity yet attained has been 424 feet per second ; this 
is in a solid cast-iron rim with numerous steel spokes. 
The average bursting velocity for solid cast rims with 
cast spokes is 400 feet per second. 

Wheels with jointed rims burst at speeds varying 
from 190 to 250 feet per second, according to the style 
of joint and its location. The following general con- 
clusions seem justified by these tests. 

1. Fly-wheels with solid rims, of the proportions 
usual among engine builders and having the usual 
number of arms, have a sufficient factor of safety at a 
rim speed of 100 feet per second if the iron is of good 
quality and there are no serious cooling strains. 

In such wheels the bending due to centrifugal force 
is slight, and may safely be disregarded. 

* See Trans. Am. Soc. Mech. Eng., Vol. XXIII. 



EXPERIMENTS ON FLY-WHEELS. 175 

2. Rim joints midway between the arms are a serious 
defect and reduce the factor of safety very materially. 
Such joints are as serious mistakes in design as would 
be a joint in the middle of a girder under a heavy load. 

3. Joints made in the ordinary manner, with internal 
flanges and bolts, are probably the worst that could be 
devised for this purpose. Under the most favorable 
circumstances they have only about one-fourth the 
strength of the solid rim and are particularly weak 
against bending. 

See Fig. 79, which shows the opening of such a joint 
and the bending of the bolts. 

In several joints of this character, on large fly- 
wheels, calculation has shown a strength less than one- 
fifth that of the rim. 

4. The type of joint known as the link or prisoner 
joint is probably the best that could be devised for 
narrow rimmed wheels not intended to carry belts, and 
possesses, when properly designed, a strength about 
two-thirds that of the solid rim. 

In 1902-04 experiments on four- foot pulle} T s were 
conducted by the writer, and the results published.* 

A cast-iron, whole rim pulley 48 inches in diameter, 
burst at 1100 rev. per min. or a linear speed of 230 ft. 
per sec, the rupture being caused by a balance weight 
of 3-J- pounds which had been riveted inside the rim by 
the makers. The centrifugal force of this weight at 
1100 rev. per min. was 2760 lb. 

A cast iron split pulley of the same dimensions burst 
at a speed of about 600 rev. per min., or a linear speed 
of only 125 ft. per sec. 

The failure was due to the unbalanced weight of the 



* Trans. Am. Soc. Mech., Eng., Vol. XXVI, 



176 MACHINE DESIGN. 

joint flanges and bolts which were located midway be- 
tween the arms. Such a pulley is not safe at high belt 
speeds. 

84. Wooden Pulleys. Experiments on the bursting 
strength of wooden pulleys were conducted at the Case 
School laboratories in 1902-3 under the writer's direc- 
tion.* 

These are of some interest in view of the use of this 
material for fly-wheel rims. As noted in Art. 82, the 
tensile stress in wood due to the centrifugal force is 
only -^ that of cast-iron under similar circumstances. 
Assuming the tensile strength of the wood to be 10000 
lbs. per sq. in., and substituting this value in the equa- 
te 2 
tion S = -T-T7- we have the bursting speed of a wooden 

pulley t;=1000 ft. per sec. nearly. 

This for wood without joints. 

The 24 inch pulleys tested had wood rims glued up in 
the usual manner and jointed at two opposite points. 
The wheels burst at speeds varying from 1700 to 2450 
rev. per min., or linear rim speeds varying from 178 to 
257 ft. per sec, thus comparing favorably with cast 
iron split pulleys. The rims usually failed at the 
points where the arms were mortised in, and the stif- 
fening braces at these points did more harm than good. 
A wooden pulley with solid rim and web remained in- 
tact at 4450 rev. per min., or 467 ft. per sec, a higher 
speed than that of any cast-iron pulley tried. 

85. Rims of Cast-iron Gears. A toothed wheel will 
burst at a less speed than a pulley because the teeth 

* Machinery, N. Y., Aug., 1905. 



ROTATING DISCS. 177 

increase the weight and therefore the centrifugal force 
without adding to the strength. 

The centrifugal force and therefore the stresses due 
to the force will be increased nearly in the ratio that 
the weight of rim and teeth is greater than the weight 
of rim alone. 

This ratio in ordinary gearing varies from 1.5 to 
1.7. We will assume 1.6 as an average value. Neg- 
lecting bending we now have from equation (89) 

9 9 



lit 

v=\l9. 



and t;=\19.2w 

= 326.2 ft. per second . . . (96) 
Including bending 

s ' =1 -KS+ro) w 

As the transverse strength of cast iron by experi- 
ment is about double the tensile strength, a larger 
value of S may be allowed in formulas (93) (94) and 
(97.) 

In built up wheels it is better to have the joints come 
near the arms to prevent the tendency of the bending 
to open the joints, and the fastenings should have the 
same tensile strength as the rim of the wheel. 

86. Rotating Discs. The formulas derived in Art. 
82 will only apply in the case of thin rims and cannot 
be used for discs or for rims having any considerable 
depth. The determination of the stresses in a rotating 
disc is a complicated and difficult problem, if the ma- 
terial is regarded as perfectly elastic. 

A rational solution of this problem may be found in 
Stodola's Steam Turbines, pp. 157-69. For the pur- 

12 



178 



MACHINE DESIGN. 



poses of this treatise an approximate solution is pre- 
ferred, the elasticity of the metal being neglected. 
This method of treatment is much simpler, and as the 
metals used are imperfectly elastic (especially the cast 
metals) the results obtained will probably be as reliable 
as any — for "practical use. 

The following discussion is an abstract of one given 
by Mr. A. M. Levin in the American Machinist * the 
notation being changed somewhat. 




Fig. 80. 



87. Plain Discs. 

Let Fig. 80 represent 
a ring of uniform 
thickness t, having 
an external diameter 
D and an internal 
diameter d, all in 
inches. 

Let v= external ve- 
locity in feet per sec- 
ond. 



Let a. 



angular velocity =-77- 



r= radius to center of gravity of half ring 

in feet. 
w= weight of metal per cubic inch. 
The value of r for a half-ring is easily proved to be : 

_2_ D z -d* 

3tt 



or 



£> 2 - 

1 

18tt 



d 2 
D 3 



in inches 



D 2 -d 



% in feet. 



* American Machinist, Oct. 20, 1904, 



ROTATING DISCS. 179 

The weight of the half -ring is : 

W=l(D 2 -cl 2 )tw 

o 

and its centrifugal force : 

n Wa 2 r aHw^—d?) , QQ s 

c= ~r = isr~ m 

Substituting for a its value in terms of v : 

c _ Uwv\JJ-d z ) / g9 x 

Now if we assume the stress on the area at AB due 
to the centrifugal force to be uniformly distributed : 
(and here lies the approximation) then will the tensile 
stress on the section be 

e C ±wv\D 2 + Dd+d 2 ) , lnA v 

S =(D=Wt= gU ' * (1 ° 0) 

For a solid disc : 

s^Jf- (101) 

For a thin ring : 

S^J^f (102) 

on the same as in equation (89). 

If the metal be perfectly elastic, Stodola's formulas 

give S= as the stress near the center when d ap- 
proaches 0— or more than twice the value given in 
(101). In view of the imperfect elasticity of the metals 
used the true value will probably be between these two. 
This value should be determined by experiment. 



180 



MACHINE DESIGN. 



88. Conical Discs. Let Fig. 81 represent a ring 
whose thickness varies uniformly from the inner to the 
outer circumference and whose dimensions are as fol- 
lows : 

D= outer diameter in inches. 

d= inner diameter " 

b= breadth of ring at inner circumference. 

m= tangent of angle of slant CAD. 



Then m 



D-d 



or b 



D-d 



b m 

By cutting the ring into slices perpendicular to the 




Fig. 81. 

axis, finding the centrifugal force for each slice and 
then integrating between D and cZ, the centrifugal 
force of the half-ring is found to be : 



c _ wv 2 (D i +Zd i -4:Dd*) 
~~ mgD 2 



(103) 



The area on the line A B to resist the centrifugal 

%tf-±Dd*) (104) 



f . (D-d) 2 A o ZwifiD* 

force is : ^ ' and S= ^-rw/ 

2m gD\D—d) 



BURSTING SPEEDS. 181 

Whend=0 : 

SJ^- (105) 

'9 

or a stress one-half that of a plain flat disc. 

89. Discs with Logarithmic Profile. A form of 
disc sometimes used for steam turbines consists of a 
solid of revolution generated by a curve of the 
equation 

y= a log I 

revolving around the x— axis. 

Mr. Levin investigates two curves of this character : 

y=log x and y=2 log -^ 

and finds the stresses to be respectively : 

When a=b S = 1.5^ (106) 

9 

When a=ib S=l.tf^ (107) 

The general equation for S in this case is : 

s==m wvl.^. (108) 

and in deriving the formulas (106) and (107) D is as- 
sumed as 8a and as da respectively. 

90. Bursting Speeds. It will be seen that all the 
formulas for centrifugal stress may be reduced to the 
general form : 

S = k— .(109) 

9 

where k is a constant depending upon the shape of the 

rotating body. 



182 



MACHINE DESIGN. 



The following table gives the values of v= * I77-, the 

bursting speed of iron in feet per second, for different 
materials and different shapes. 



TABLE XXXII. 
BURSTING SPEEDS IN FEET PER SECOND. 





w 


a 
S 


Values of v. 


Metal. 


Thin 
Ring. 


Perforated 

Disc 
(Stodola). 


Flat 
Disc. 


Taper 
Disc. 


Logar- 
ithmic 
Disc. 




k=12 


fc=9 


fc=4 


k=2 


fc=1.5 


Cast Iron 

Manganese Bronze 
Soft Steel 


.026 

.0315 

.028 


O O 00 
OOO 
OOO 
OOO 


430 
715 
760 


500 

825 
880 


745 
1240 
1315 


1050 
1750 

1860 


1215 
2050 
2140 









Problems. 

1. Determine bursting speed in revolutions per minute, of a 
gear 48 inches in diameter with six arms, if the thickness of 
rim is .75 inch. 

(1) Considering centrifugal tension alone. 

(2) Including bending of rim due to centrifugal force as- 
suming that I the stress due to bending is relieved by the 
stretching of the arms. 

2. Design a link joint for the rim of a fly-wheel, the rim 
being 8 in. wide, 12 in. deep and 18 ft. mean diameter, the 
links to have a tensile strength of 65000 lb. per sq. in; 
Determine the relative strength of joint and the probable 
bursting speed, 



BURSTING SPEEDS. 183 

3. Discuss the proportions of one of the following wheels in 
the laboratory and criticise dimensions. 

(a) Fly-wheel, Allis engine. 

(b) Fly-wheel, Fairbanks gas engine. 

(c) Fly-wheel, air compressor, 
(a) Fly-wheel, Ball engine. 

(e) Fly-wheel, ammonia compressor. 

4. Determine the value of C in formula (103) by calculation. 

5. A Delaval turbine disc is made of soft steel in the shape 
of the logarithmic curve without any hole at the center. 
Determine the probable bursting speed if the disc is 8 inches in 
diameter. 

6. A wheel rim is made of cast iron in the shape of a ring 
having diameters of 4£ feet and 6 feet, inside and outside. 
Determine probable bursting speed. 

7. Substitute the value for centrifugal force in place of 
internal pressure in Barlow's formula (b) Art. 12, and derive a 

value for S in a rotating ring. Test this for d==- and compare 

2 

with formulas in preceding article. 



CHAPTER XII. 

TRANSMISSION BY BELTS AND ROPES. 

91. Friction of Belting. The transmitting power of 
a belt is due to its friction on the pulley, and this friction 
is equal to the difference between the tensions of the 
driving and slack sides of the belt. 

Let w = width of belt. 

T 1= = tension of driving side. 
T 2 = tension of slack side. 
^\ ! |\ H = friction of belt. 




/== coefficient of friction be- 
*"V>X tween belt and pulley. 

7-V = arc of contact in circu- 

Fig. 82. lar measure. 

The tension T at any part of the arc of contact is in- 
termediate between Ti and T 2 . 

Let AB Fig. 82 be an indefinitely short element of 
the arc of contact, so that the tensions at A and B 
differ only by the amount dT. 

dT will then equal the friction on AB which we may 
call dR. 

Draw the intersecting tangents OT and T to rep- 
resent the tensions and find their radial resultant 
OP. Then will OP represent the normal pressure on 
the arc AB which we will call P. 

<OTP=<ACB = dO 
.-. P=TdO 

184: 



FRICTION OF BELTING. 

The friction on AB is 

fP=fTdO 
or dT=dR=fTde 

and fdO=^ 

Integrating for the whole arc 6 : 

j-f 



185 



rn — lOQ e yp 



% 6 

E=T 1 -T 8 =r I (l-e-/ e ) ( n °) 

The average value of / for leather belts on iron pul- 
leys as determined by experiment is f=0.27. 

If we denote expression (1—e—fO) by C, then for dif- 
ferent arcs of contact C has the following values : 



Arc of 
Contact. 


90° 


110° 


130° 


150° 


180° 


210° 


240° 


C 


.345 


.404 


.458 


.506 


.571 


.627 


.676 



The friction or force transmitted by a belt per inch 
of width is then 

B=CT 1 ........ (Ill) 

and Ti must not exceed the safe working tensile 
strength of the material. 

A handy rule for calculating belts assumes 0=.5 
which means that the force which a belt will transmit 



186 MACHINE DESIGN. 

under ordinary conditions is one-half its tensile 
strength. 

92. Strength of Belting. The strength of belting 
varies widely and only average values can be given. 
According to experiments made by the author good oak 
tanned belting has a breaking strength per inch of 
width as follows : 

Single. Double. 

Solid leather 900 lb. 1400 lb. 

Where riveted 600 lb. 1200 lb. 

Where laced 350 lb. 

Canvas belting has approximately the same strength 
as leather. Tests of rubber coated canvas belts 4-ply, 
8 inches wide, show a tensile strength of from 840 lb. 
to 930 lb. per inch of width. 

93. Taylors Experiments. The experiments of Mr. 
F. W. Taylor, as reported by him in Trans. Am. Soc. 
Mech. Eng. Yol. XV. afford the most valuable data 
now available on the performance of belts in actual 
service. 

These experiments were carried on during a period 
of nine years at the Midvale Steel Works. Mr. Taylor's 
conclusions may be epitomized as follows : 

1. Narrow double belts are more economical than 
single ones of a greater width. 

2. All joints should be spliced and cemented. 

3. The most economical belt speed is from 4000 to 
4500 ft. per min. 

4. The working tension of a double belt should not 
exceed 35 lb. per inch of width, but the belt may be 
first tightened to about double this. 



RULES FOR WIDTH OF BELTS. 187 

5. Belts should be cleaned and greased every six 
months. 

6. The best length is from 20 to 25 feet between 
centers. 

94. Rules for Width of Belts. It will be noticed 
that Mr. Taylor recommends a working tension only 
to to ¥ V the breaking strength of the belt. He justifies 
this by saying that belts so designed gave much less 
trouble from stoppage and repairs and were conse- 
quently more economical than those designed by the 
ordinary rules. 

In the following formulas 50 lb. per inch of width 
is allowed for double belts and 30 lb. for single belts. 
These are suitable values for belts which are not run- 
ning continuously. The formulas may be easily 
changed for other thicknesses and for other values of 
CT X . 

Let HP— horse power transmitted. 

D= diameter of driving pulley in inches. 
iV=no. rev. per min. of pulley. 

The moment of force transmitted by belt is 

RD CT.wD T 

2 _ 2 

aud HP=-™-= O^wDN . _ . (m) 

63025 126050 V ' 

Substituting the values assumed for CT t and solving 
for w : 

Single belts w=4Q00ji£ (113) 

Double belts w=2500jL£ (114) 



188 MACHINE DESIGN. 

The most convenient rules for belting are those 
which give the horse-power of a belt in terms of the 
surface passing a fixed point per minute. 

In formula (lis) nP-<*&™ 

we will substitute the following : 

IT = width of belt in feet ~ 

V = velocity in ft. per min. = * 

Hp= lUCT y WV 

126050*- 

Substituting values of C and T x as before and 
solving for WV= square feet per minute we have ap- 
proximately : 

Single belts WV=90HP. . . . ,(115) 
Double belts WV=5&HP. . . . .(116) 

95. Speed of Belting. As in the case of pulley 
rims, so in that of belts a certain amount of tension is 
caused by the centrifugal force of the belt as it passes 
around the pulley. 

From equation (89) S = ■ 

where v= velocity in ft. per sec. 

w= weight of material per cu. in. 
$ = tensile stress per sq. in. 

To make this formula more convenient for use we 
will make the following changes in the constants : 



RULES FOR WIDTH OF BELTS. 189 

Let V= velocity of belt in ft. per minute = 60^. 
w= weight of ordinary belting. 

= .032 per cu. in. 
S t = tensile stress per inch width, caused by 
centrifugal force. 
= about T \ S for single belts. 

Then t,-£ 

Substituting these values in (89) and solving for S 1 

^ 1== 1610000 ("^ 

The speed usually given as a safe limit for ordinary 
belts is 3000 ft. per min., but belts are sometimes run 
at a speed exceeding 6000 ft. per min. 

Substituting different values of Fin the formula we 
have : 

F=3000 # 1= 5.59 1b. 

F=4000 S,= 9.94 1b. 

F=5000 ^=15.53 lb. 

F=6000 £ x =22.36 lb. 

The values of S x for double belts will be nearly 

twice those given above. At a speed of 5000 ft. per 

minute the maximum tension per inch of width on a 

single belt designed by formula (113), if we call 

C — . 5, will be : 

(30X2) + 15. = 75 lb. 

giving a factor of safety of eight or ten at the splices. 
In a similar manner we find the maximum tension 
per inch of width of a double belt to be : 

(50X2) + 30 =130 lb. 



190 MACHINE DESIGN. 

and the margin of safety about the same as in single 
belting. 

A double belt is stiffer than a single one and should 
not be used on pulleys less than one foot in diameter. 
Triple belts can be used successfully on pulleys over 20 
inches in diameter. 

96. Manila Rope Transmission. Eopes are some- 
times used instead of flat belts for transmitting power 
short distances. They possess the following advan- 
tages : they are cheaper than belts in first cost ; they 
are flexible in every direction and can be carried 
around corners readily. They have however the dis- 
advantage of being less efficient in transmission than 
leather belts and less durable ; they are also some- 
what difficult to splice or repair. 

There are two systems of rope driving in common 
use : the English and the American. In the former 
there are as many separate ropes as there are grooves 
in one pulley, each rope being an endless loop always 
running in one groove. 

In the American system one continuous rope is used 
passing back and forth from one groove to another 
and finally returning to the starting point. 

The advantage of the English system consists in 
the fact that one of the ropes may fail without causing 
a breakdown of the entire drive, there usually being 
two or three ropes in excess of the number actually 
necessary. On the other hand the American system 
has the advantage of a uniform regulation of the ten- 
sion on all the plies of rope. The guide pulley, which 
guides the last slack turn of rope back to the starting 
point, is usually also a tension pulley and can be 
weighted to secure any desired tension. The English 



STRENGTH OF MANILA ROPES. 191 

method is most used for heavy drives from engines to 
head shafts ; the American for lighter work in dis- 
tributing power to the different rooms of a factory. 
The grooves in the pulleys for manila or cotton ropes 
usually have their sides inclined at an angle of about 
45°, thus wedging the rope in the groove. 

The Walker groove has curved sides as shown in 
Fig. 83, the curvature 
increasing towards the 
bottom. As the rope 
wears and stretches it 
becomes smaller and sinks 
deeper in the groove ; the 
sides of the groove being 
more oblique near the 
bottom, the older rope is 
not pinched so hard as the 
newer and this tends to 




Fig. 83. 



throw more of the work on the latter. 

97. Strength of Manila Ropes. The formulas for 
transmission by ropes are similar to those for belts, 
the values for S and <£ being changed. The ultimate 
tensile strength of manila and hemp rope is about 
10000 lb. per sq. in. 

To insure durability and efficiency it has been found 
best in practice to use a large factor of safety. Prof. 
Forrest R. Jones in his book on Machine Design 
recommends a maximum tension of 200 d 2 pounds 
where d is the diameter of rope in inches. This cor- 
responds to a tensile stress of 255 lb. per sq. in. or a 
factor of safety of about 40. 

The value of / for manila on metal is about 0.12, 
but as the normal pressure between the two surfaces 



192 MACHINE DESIGN. 

is increased by the wedge action of the rope in the 
groove we shall have the apparent value of / : 

f 1 = f -^sin-^-where 

a = angle of groove, 

For a = 45° to 30° 

f 1 varies from 0.3 to 0.5 and these values should be 
used in formula (110). 

(1— e~ J ) in this formula, for an arc of contact of 
150°, becomes either .54 or .73 according as/ 1 is taken 
0.3 or 0.5. 

If T x is assumed as 250 lb. per sq. in. , the force R 
transmitted by the rope varies from 135 lb. to 185 
lb. per sq. in. area of rope section. 

The following table gives the horse-power of manila 
ropes based on a maximum tension of 255 lb. per sq. in. 

TABLE XXXIII'. 

Table of the horse-power of transmission rope, reprinted 
from the transactions of the American Society of Mechanical 
Engineers, Vol. 12, page 230, Article on " Eope Driving" by 
C. W. Hunt. 

The working strain is 800 lb. for a 2-inch diameter rope and 
is the same at all speeds, due allowance having been made for 
loss by centrifugal force. 



WIRE ROPE TRANSMISSION. 



193 





SPEED OF THE ROPE IN FEET PER MINUTE. 


Lest 
. Pul- 
Ins. 


Diam 

Rope 

In 


1500 


2000 


2500 


3000 


3500 


4000 


4500 


5000 


6000 


7000 


si" 


f 

7 
8 

1 
U 

n 
if 

2 


3.3 

4.5 

5.8 

9.2 

13.1 

18.0 

23.1 


4.3 
5.9 

7.7 
12.1 
17.4 
23.7 
30.8 


5.2 

7.0 
9.2 
14.3 
20.7 
28.2 
36.8 


5.8 
8.2 
10.7 
16.8 
23.1 
32.8 
42.8 


6.7 
9.1 
11.9 
18.6 
26.8 
36.4 
47.6 


7.2 
9.8 
12.8 
20.0 
28.8 
39.2 
51.2 


7.7 
10.8 
13.6 
21.2 
30.6 
41.5 
54.4 


7.7 
10.8 
13.7 
21:4 
30.8 
41.8 
54.8 


7.1 
9.3 
12.5 
19.5 
28.2 
37.4 
50.0 


4.9 
6.9 
8.8 
13.8 
19.8 
27.6 
35.2 


30 
36 

42 
54 
60 

72 
84 



98. Wire Rope Transmission. Wire ropes have 
been used to transmit power where the distances were 
too great for belting or hemp rope transmission. The 
increased use of electrical transmission is gradually 
crowding out this latter form of rope driving. 

For comparatively short distances 
of from 100 to 500 yards wire rope still 
offers a cheap and simple means of 
carrying power. 

The pulleys or wheels are entirely 
different from those used with manila 
ropes. 

Fig. 84 shows a section of the rim 
of such a pulley. The rope does not 
touch the sides of the groove but 
rests on a shallow depression in a 
wooden, leather or rubber filling at the bottom. The 
high side flanges prevent the rope from leaving the 
pulley when swaying on account of the high speed. 

The pulleys must be large, usually about 100 times 
13 




Fig. 84. 



194 



MACHINE DESIGN. 



the diameter of rope used, and run at comparatively 
high speeds. The ropes should not be less than 200 
feet long unless some form of tightening pulley is used. 
—Table XXXIV. is taken from Eoebling. 

Long ropes should be supported by idle pulleys every 
400 feet. 

TABLE XXXIV. 

TRANSMISSION OF POWER BY WIRE ROPE. 

Showing necessary size and speed of wheels and rope to obtain 
any desired amount of power. 





03 

o S 




© 








© 


is 

5£ 


° o 

■*! 

i> 

s © 


-u © 
© ft 

Q<3 


o 

o 

w 


'St 


o o 

11 

II 


ft 

4S © 
© ft 

i a 

5 'S 


o 
o 


4 


80 


5-8 


3.3 


10 


80 


11-16 


58.4 




100 


5-8 


4.1 




100 


11-16 


73. 




120 


5-8 


5. 




120 


11-16 


87.6 




140 


5.8 


5.8 




140 


11-16 


102.2 


5 


80 


7-16 


6.9 


11 


80 


11-16 


75.5 




100 


7-16 


8.6 




100 


11-16 


94.4 




120 


7-16 


10.3 




120 


11-16 


113.3 




140 


7-16 


12.1 




140 


11-16 


132.1 


6 


80 


1-2 


10.7 


12 


80 


3-4 


99.3 




100 


1-2 


13.4 




100 


3-4 


124.1 




120 


1-2 


16.1 




120 


3-4 


148.9 




140 


1-2 


18.7 




140 


3-4 


173.7 


7 


80 


9-16 


16.9 


13 


80 


3-4 


122.6 




100 


9-16 


21.1 




100 


3-4 


153.2 




120 


9-16 


25.3 




120 


3-4 


183.9 


8 


80 


5-8 


22. 


14 


80 


7-8 


148. 




100 


5-8 


27.5 




100 


7-8 


185. 




120 


5-8 


33.0 




120 


7-8 


222. 


9 


80 


5-8 


41.5 


15 


80 


7-8 


217. 




100 


5-8 


51.9 




100 


7-8 


259. 




120 


5-8 


62.2 




120 


7-8 


300. 



WIRE ROPE TRANSMISSION. 195 

PROBLEMS. 

1. Design a main driving belt to transmit 150 HP. from a 
belt wheel 18 ft. in diameter and making 80 rev. per min. 
The belt to be double leather without rivets. 

2. Investigate driving belt on Allis engine and calculate the 
horse-power it is capable of transmitting economically. 

3. Calculate the total maximum tension per inch of width 
due to load and to centrifugal force of the driving belt on the 
motor used for driving machine shop, assuming the maximum 
load to be 10 HP. 

4. Design a manila rope drive, English system, to transmit 

500 HP. , the wheel on the engine being 20 feet, in diameter 

and making 60 rev. per min. Use Hunt's table and then 

check by calculating the centrifugal tension and the total 

v 2 
maximum tension on each rope. Assume S= ™ where v= 

feet per second. 

5. Design a wire rope transmission to carry 120 HP. a dis- 
tance of one-quarter mile using two ropes. Determine working 
and maximum tension on rope, length of rope, diameter and 
speed of pulleys and number of supporting pulleys. 



INDEX. 



ART. PAGE. 

Abbreviations 2 1 

Adjustment op Bearings 43 96 

Alloys 3 4 

Arms of Pulleys 77 157 

Ball Bearings, 

In general 57 118 

Conical 58 119 

Cylindrical 58 118 

Design of 61 123 

Materials of 60 122 

Step or thrust ' 59 120 

Beams, formulas for 5 12 

Of uniform strength 6 12 

Bearings, 

Adjustment of 43 96 

Ball 57 118 

Cylindrical 42 96 

Engine 43 97 

Lathe.. 43 98 

Lubrication of 44 99 

Roller 62 123 

Sliding 36 86 

Step or thrust 51 111 

Thrust 56 116 

Belting, 

Centrifugal tension of 95 189 

Friction of 91 184 

Speed of 95 188 

Strength of 92 186 

Taylor's experiments on 93 186 

Width of . .• 94 187 

Boiler Shells 11 25 

Tubes 14 38 

197 



198 MACHINE DESIGN. 

ART. PAGE. 

Bolts, 

Coupling.... : 69 138 

Dimensions of 18 54 

Eye or hook "... 20 57 

Bronzes 3 5 

Butt Joints 23 62 

Cabinet Supports 9 19 

Caps and Bolts 50 109 

Cast Iron 3 3 

Centrifugal Oilers 44 101 

Chain Driving, 

Block 78 159 

Roller 78 160 

Silent 79 161 

Clutches, 

Conical 68 137 

Roller 68 138 

Weston 68 136 

Collar Bearing 53 113 

Compound 56 116 

Column Formulas 5 9 

Constants, 

Columns . ... 5 10 

Cross-sections 5 11 

Cotters 29 69 

Couplings, 

Bolts for 69 138 

Clutch 68 135 

Flange 67 133 

Muff 67 135 

Sleeve 67 134 

Cranks 80 163 

Crank Pins, 

Heating of 47 104 

Pressure on , 46 103 

Cross-sections , 10 21 

Cylinders, 

Hydraulic 12 27 

Steam...... •'.. 16 42 

Design, General Principles op 8 15 

Discs, Rotating 86 177 



INDEX. 199 



ART. 



Disc, — Continued. 

Conical 88 180 

Logarithmic '. . . , 89 181 

Plain 87 178 

Speeds of 90 181 

Factors of Safety 7 13 

Flat Plates, 

Formulas for 17 48 

Tests of 17 51 

FlyWheels 81 166 

Experiments on , 83 170 

Rim Joints of 83 175 

Safe Speed of 82 167 

Flues, Strength of 14 38 

Formulas, General 5 8 

Frame Design 10 20 

Friction, 

Belts. 91 184 

Journals 45 102 

Experiments on 48 107 

Pivots 52 112 

Scbiele pivot 55 114 

Gears, 

Arms of 77 157 

Rims of 77 157 

Teeth of 72 146 

Gibs 28 72 

" 38 89 

Guides, Circular 40 91 

Hangers 71 141 

Heating of Journals , 47 107 

Hooks, Design of 20 57 

Iron, 

Cast 3 3 

Malleable 3 4 

Wrought 3 2 

Joints, 

Butt 23 62 

Lap 22 61 

Riveted 21 58 

Joint Pins 28 69 



200 MACHINE DESIGN. 

ART. PAGE. 

Journals 42 96 

Experiments on 48 107 

Friction of ... 45 102 

Heating of 47 104 

Pressure on 46 103 

Strength of 49 108 

Keys, 

Cotter 28 69 

Shafting 70 139 

Woodruff 70 141 

Lap Joints 22 61 

Legs of Machines 9 19 

Levers 80 163 

Lubrication of Bearings 44 99 

Machine Frames 10 20 

Malleable Iron 3 4 

Materials of Construction 3 2 

Notation 4 8 

Oil Cups 44 100 

Packings for Glands 41 92 

Pipe, 

Sizes 13 30 

Fittings 15 40 

Pivots, 

Conical 54 113 

Flat 52 112 

Schiele 55 114 

Plates, 

Flat 17 48 

Narrow 27 68 

Pressure on Journals 46 103 

Pulleys, 

Arms of 77 157 

Cast Iron 83 175 

Wooden , 84 176 

Ring Oiler 44 101 

Riveted Joints 21 58 

Efficiency of 24 62 

Narrow Plates 27 68 

Practical rules for 26 64 

Special forms of 25 63 



INDEX. 201 



Roller Bearings, 

Conical 63 124 

Cylindrical 62 123 

Hyatt 64 125 

Step 65 127 

Tests of 64 126 

Rope, Transmission, 

Manila 96 190 

Strength of 97 191 

Wire 98 193 

Schiele Pivot 55 114 

Screws, Machine 19 57 

Shafting, 

Diameter of 66 132 

Keys for 70 139 

Spanof 66 133 

Strength of 66 130 

Shells, Strength of, 

Thick 12 27 

Thin 11 25 

Slides, 

Angular 37 87 

Flat 39 89 

Gibbed , 38 88 

Springs, 

Elliptic 35 83 

Flat.,... 34 81 

Helical 30 73 

Square wire 31 75 

Testsof 32 76 

Torsion 33 79 

Steam Cylinders, 

Strength of 16 42 

Testsof 16 45 

Steel 3 3 

Strength of Metals, 

Cast , , 3 7 

Wrought 3 6 

Stuffing Boxes 41 91 

Supports, Machine 9 18 



202 MACHINE DESIGN. 

ART. PAGE. 

Teeth of Gears, 

Bevel 76 155 

Cut 73 150 

Experiments on 75 153 

Lewis' formula for 74 151 

Proportions of , 72 148 

Strength of 73 148 

Velocityof 74 152 

Thrust Bearing 56 116 

Tubes, Boiler 14 38 

Units and Definitions 1 1 

Wrought Iron 3 2 



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